Start with a set X of "states" and a probability measure mu on X. So mu is a
rule that assigns a number between 0 and 1 to every measurable subset, mu(X)
= 1, and mu is countable-additive: if pieces are disjoint, their
probabilities add.
Now take a bijection T of X onto itself that preserves mu: mu(T^-1(A)) =
mu(A) for every measurable A. That is, T is a MEASURE-PRESERVING
TRANSFORMATION (abbreviated mpt). It is reversible (T has an inverse), it is
time, and mu is a stationary distribution.
Example. Let X be a coin and mu be the fair coin measure. Then T is the flip
map that switches heads to tails and back. Trivial but valid.
Example. Let X be the unit interval [0,1) with Lebesgue length as mu. Let
T(x) be the fractional part of 10x, i.e. T(x) = 10x - floor(10x). This is
the "decimal shift" map. It preserves Lebesgue length. It is invertible when
two-sided decimal expansions are used.
Every statistical-mechanical system with a stationary distribution is of
this form: X = set of states, T = one step of time, mu = the equilibrium
measure.
The SMOOTH REALIZATION PROBLEM asks:
Which mpts can be drawn using ordinary geometry?
"Ordinary geometry" means: X is (or is measurably bijective to) a compact
smooth manifold M of finite dimension d; mu is (or is pushforward to) a
smooth positive probability volume; T corresponds to a C^infinity
diffeomorphism S of M preserving that volume.
DIFFEOMORPHISM: a bijection M -> M that is C^infinity and whose inverse is
also C^infinity. So it is smooth, invertible, and its inverse is smooth too.
DIRECTIONS ON A MANIFOLD: at every point x of M, the space of tangent
directions is a d-dimensional real vector space called the tangent space at
x, written T_x M. The derivative of a diffeomorphism S at x is a linear map
D S_x : T_x M -> T_{S x} M. This is the linear approximation of S near x.
SMOOTH POSITIVE DENSITY (of a probability measure on M): nu is called smooth
positive volume if there is a C^infinity function f: M -> (0, infinity) with
integral 1 over M relative to the ordinary volume form |dx_1 dx_2 ... dx_d|
in any chart, and
d nu = f(x) |dx_1 dx_2 ... dx_d|.
Equivalently, nu has density f which is everywhere positive and smooth. The
function f is called the DENSITY of nu. Ordinary volume itself is the
special case f == 1.
Note: the manifold may have a smooth boundary, and no orientation is needed
for anything stated. Boundary and nonorientable cases are handled inline.
A STANDARD NONATOMIC PROBABILITY SPACE is any probability space that is
isomorphic (up to null sets) to the unit interval with Lebesgue measure. A
"nonatomic" measure has no atoms: no single point carries positive mass. The
unit interval with Lebesgue measure is the model example, and measurable
isomorphism theory says any two nonatomic standard spaces are the same up to
null sets. To say "X is a standard nonatomic probability space" means: for
the purposes of this problem, X looks like Lebesgue measure on [0,1].
WHAT ISOMORPHISM MEANS FOR SYSTEMS. Two mpts (X, mu, T) and (Y, nu, S) are
ISOMORPHIC (or measurably conjugate) if there exist invariant conull
measurable subsets X0 of X and Y0 of Y, and a measurable bijection F : X0 ->
Y0 with measurable inverse such that
F o T = S o F (as functions on X0),
and F carries mu (restricted to X0) to nu (restricted to Y0). No continuity
of F is required. Conjugacy is the notion of sameness for mpts.
So the question is: does every mpt (X, mu, T) of a standard nonatomic space
have a smooth positive-volume model in the sense just defined? The answer is
negative.
Observation of the system asks "which bin is it in?" from a finite partition
P of X into r pieces P_1, ..., P_r. The answer is a random variable with r
possible values, probabilities p_1, ..., p_r. The SHANNON ENTROPY of that
answer is
H(P) = - sum_i p_i log_2 p_i (with 0 log_2 0 := 0).
H(P) is measured in bits. It is largest when all p_i are equal and equal to
log_2 r. It is 0 when P is degenerate (one atom has probability 1).
Now consider a second observation. Partition X by time-1 atoms T^-1(P). That
says "which atom did the system come from at time -1?" Together, P and
T^-1(P) have a joint answer described by a joint distribution on r x r
outcomes. Iterate: at times 0, 1, ..., n-1, an r-symbol pattern appears. The
empirical laws of these n-step patterns, as n grows, generate entropy at
some rate. That rate is
h(T, P) = lim_{n -> infinity} (1/n) H( P v T^-1 P v ... v T^{-(n-1)} P ),
where "v" is join (the coarsest partition refining both). The limit exists
by subadditivity: the sequence H_n = H(P v ... v T^{-(n-1)}P) satisfies
H_{n+m} <= H_n + H_m, and Fekete's lemma gives existence of the limit as a
real number in [0, log r].
Then the KOLMOGOROV-SINAI ENTROPY OF T is
h(T) = sup_P h(T, P),
the supremum over all finite measurable partitions. It is finite when any
finite-entropy partition generates, and it is a conjugacy invariant: two
isomorphic systems have the same h.
INTUITION. h(T) is the number of bits per time step gained by watching the
system at arbitrarily fine detail. An identically zero map has h = 0. An
independent coin flip at each time has h = 1 bit per step. A rotation is
zero-entropy: on any fine enough grid, the pattern of atoms appearing at
times 0, ..., n-1 only grows polynomially in n, so per-step entropy goes to
zero. A Bernoulli shift on two symbols (each step is an independent fair
coin) has h = 1.
ENTROPY OF A FINITE RANDOM VARIABLE. If Y is a random variable taking
finitely many values y_1, ..., y_r with probabilities p_1, ..., p_r, then
H(Y) is H of its law. So H(Y) = - sum p_i log_2 p_i. Same notation.
CONDITIONAL ENTROPY. If Y and Z are finite random variables,
H(Y | Z) = sum_{z : Pr(Z = z) > 0} Pr(Z = z) H( law of Y | Z = z ).
This is "entropy of Y, given Z." If Y is independent of Z, H(Y | Z) = H(Y).
If Y is determined by Z, H(Y | Z) = 0. Always H(Y, Z) = H(Z) + H(Y | Z) and
H(Y | Z) <= H(Y).
WHY THIS MATTERS FOR SMOOTH MODELS. Ruelle's entropy inequality. If S : M ->
M is a C^1 diffeomorphism of a compact d-dimensional manifold, if the
derivative DS is bounded in norm, and if nu is any S-invariant probability
on M, then
h_nu(S) <= sum of positive Lyapunov exponents of S with respect to nu,
and the right side is finite. Ruelle 1978, building on Oseledets 1968. So
any smooth model has h finite. This is the FIRST restriction on which
systems can have smooth positive-volume models: h must be finite. Ruelle
states the result as
"an upper bound to the measure-theoretic entropy h(p) of any
probability measure p invariant under a differentiable map f of a
compact manifold M into itself, in terms of characteristic exponents
introduced by the non-commutative ergodic theorem of Oseledec."
./katokrefs/ruelle1978.pdf
./oseledets-1968-multiplicative-ergodic.pdf
Katok and Thouvenot, and work following them, showed finite entropy
is not sufficient in a certain sense. The example below shows finite entropy
is not sufficient even when the density is smooth and positive: not every
finite-entropy system has such a model.
Zero entropy is the extreme case of finite entropy. The constructed example
has zero entropy. So even the easiest systems, ones that produce almost no
information per step, can fail to have smooth positive-volume models.
THEOREM (main theorem). There is an ergodic invertible mpt (X, mu, T) of a
standard nonatomic probability space such that:
(i) h_mu(T) = 0.
(ii) There is NO measurably-conjugate system of the following form:
M a compact finite-dimensional C^infinity manifold (possibly with
smooth boundary, possibly nonorientable), nu a smooth positive
probability density on M, S a C^infinity diffeomorphism of M preserving
nu.
In words: the construction produces a single zero-entropy ergodic system. No
matter what dimension considered, no matter what compact manifold, no matter
what smooth positive density on it, no matter what smooth diffeomorphism
preserving that density, the system can never be isomorphic to it.
The dimension and manifold are not chosen in advance, and are not chosen for
each counterexample after the fact: the SYSTEM (X, mu, T) is fixed once and
for all, then a model of ANY dimension is ruled out. This is important: a
weaker statement would say for each dimension d there is a system that has
no model of dimension d. That would be easier. The proof gives a single
system has no model in any dimension.
The manifold being allowed to have boundary and to be nonorientable makes
the counterexample stronger, because it closes more doors.
"ERGODIC." An invariant set of the mpt is a measurable A with T^-1(A) = A up
to null sets. Trivial examples are null sets and conulls. A system is
ergodic if all invariant sets are trivial. Equivalently, if mu is an extreme
point of the convex set of invariant probabilities. In the constructed
example ergodicity matters because the zero-entropy argument for the whole
system needs a single block to behave statistically, which ergodicity
guarantees.
"ATOMLESS" / "STANDARD." As above: null set-isomorphic to [0,1] with
Lebesgue measure. Every complete separable metric space with a nonatomic
probability is standard; it is the umbrella property under which ergodic
theory "forgets fidgety set-theory decoration."
Four prior theorems about realization of mpts are the ancestors of this work
and explain why the new counterexample is not just a small improvement.
Lind and Thouvenot (1977) proved that every ergodic invertible mpt of a
standard nonatomic probability space with finite entropy is isomorphic to a
Lebesgue-measure- preserving HOMEOMORPHISM of the two-torus. Homeomorphism
means continuous and bijective with continuous inverse -- not
differentiable, not C^1.
What this says: it is always possible to realize every finite-entropy system
on a compact manifold, with a transformation that is only continuous and an
invariant measure that is ordinary Lebesgue-measure-density-1 on the torus.
Lebesgue on the torus IS a smooth positive density, so the density part is
fine. The gap is the smoothness of the map.
Quas and Soo (2016) extended Lind-Thouvenot. If A is a hyperbolic toral
automorphism whose characteristic polynomial has no root of unity as root,
then A (or any sufficiently high iterate A^N) represents EVERY ergodic mpt
of finite entropy, with measure entropy strictly smaller than the
topological entropy of A, using an A-invariant BOREL PROBABILITY OF FULL
SUPPORT on the torus. Full support means every open set has positive
measure. But the invariant Borel probability can be SINGULAR with respect to
Lebesgue: it can live on a set of Lebesgue measure zero. A singular measure
is not a smooth positive density. ./quas-soo-2016-uvic.pdf
Katok and Thouvenot (1997) constructed ergodic Z^2 actions which have no
LIPSCHITZ realization on a compact space of finite box dimension, even when
the model may carry any invariant Borel probability. This is about ACTIONS
OF Z^2 (two commuting generators realized simultaneously), not about a
single transformation. The obstruction does not answer the problem for one
transformation. ./katok-thouvenot-1997-aihpb.pdf
Earlier anti-classification machinery is due to Foreman, Rudolph, and Weiss
(2011); a subsequent development for torus diffeomorphisms appears in
Foreman and Weiss (2022), Kunde (2024), and Gerber and Kunde (2025). These
proved anti-classification results: the relation "is measurably isomorphic
to" is not Borel on certain classes of smooth transformations. Foreman-Weiss
2022 uses an invariant measure merely EQUIVALENT to volume, not necessarily
with smooth positive density with smooth positive function. These results
show that classifying smooth systems is hard, but the results do NOT say
that a specific abstract zero-entropy system lacks a smooth model. Two
different questions. ./foreman-weiss-2022-torus-unclassifiable.pdf
./gerber-kunde-2025-2509.09003-mixing-zero-entropy.pdf
The result that follows is therefore new:
The first example of a single ergodic zero-entropy system with NO smooth
positive-DENSITY model on any compact finite-dimensional manifold, where the
density is smooth and strictly positive and the map is C^infinity.
Every one of the four closest results above leaves a gap that the proof
closes. Lind-Thouvenot gives homeomorphisms. Quas-Soo gives singular
measures. Katok-Thouvenot gives actions of Z^2. Foreman-Weiss and Kunde give
anti-classification results inside families of smooth systems.
A full-support measure can be extremely singular: it can be a single orbit's
empirical average spread over a fractal set of measure zero (respecting full
support topologically). Think of the map: pick one point x, look at its
empirical measure along the orbit, and it can be weird. A full-support
invariant measure on the torus can easily be singular with respect to
Lebesgue.
A smooth positive density is different in a very strong way: it is
ABSOLUTELY CONTINUOUS with respect to ordinary volume, with density bounded
above and below by positive finite constants on any compact submanifold. It
has no fractal support. It is the closest notion to "uniformly spread out".
Why this matters for the switch construction: at some point in the proof, a
JACOBIAN is computed. A Jacobian is the local volume multiplier of a smooth
map. "1 + o(1) Jacobian" means the map multiplies volumes by 1 plus a
vanishing error. To turn "the map multiplies volumes by 1+o(1)" into "the
output measure is at most 2 times the input measure" the input measure is
needed to have a density that is smooth and positive, so that comparing
volumes at two nearby points is controlled. If the measure were singular, a
map could have Jacobian 1 and still concentrate measure by arbitrary
factors. If the density were merely L^1 or continuous, the Jacobian argument
would be less robust.
Two consequences follow:
(i) The switch inequality (z_k)_*(rho_k on G_k) <= 2 nu is only asserted
for smooth positive densities.
(ii) The theorem does NOT claim that a full-support singular invariant
measure is impossible. In fact Quas-Soo shows the opposite.
So the proof concerns the smooth positive density model specifically. Its
counterexample is only consistent with the prior positive results because
each of them relaxes one of the smoothness, positivity, or single-
transformation restrictions.
The next construction is central. It is what the combination of smoothness
and positive density provides, and it is what the
zero-entropy example below is designed to conflict with.
SETUP: a compact d-manifold M is fixed, with a smooth positive probability
volume nu (density f > 0, smooth), a C^infinity diffeomorphism S preserving
nu. The times are n = 2^k, dyadic. At each scale k the construction builds a
certain "switch map" z_k.
STEP 1: PARTITION INTO CELLS. Take a partition of M into measurable cells,
at most exp(C B_k n) of them, where C is a fixed constant depending only on
the manifold, dimension, and a fixed tolerance. The sequence B_k has two key
properties:
- sum_k B_k^16 < infinity (this defeats any symbolic system
that only offers tests at widely-spaced scales);
- at each dyadic scale k, B_k 2^k is large compared to log(2^k) = k
(this lets the derivative estimates work at that scale).
The specific recipe is B_k = b_k^{1/8}, where b_k comes from the stationary
stationary estimate and satisfies b_k >= 2^{-k/10} = n^{-1/10} and sum_k
b_k^2 < infinity. Then B_k >= n^{-1/80}, so B_k n >= n^{79/80}, and sum_k
B_k^16 = sum_k b_k^2 < infinity. So the cell count exp(C B_k n) is huge but
subexponential in n scaled with B_k -> 0.
STEP 2: SAMPLE A PAIR OF POINTS IN A CELL. Build the distribution rho_k on M
x M as follows. First pick a cell Q with probability nu(Q). Then pick two
independent points x, y uniformly (by nu conditioned on Q) inside that cell.
Equivalently:
rho_k = sum_{Q} ( (nu restricted to Q) x (nu restricted to Q) ) / nu(Q).
This is a probability on M x M. Both marginals are exactly nu.
STEP 3: THE SWITCH MAP z. On a large measurable subset G_k of M x M (with
rho_k(G_k) >= 1 - eta, eta being a fixed tolerance like 0.01), there is a
measurable map z : G_k -> M such that:
(a) SHADOWING: For (x,y) in G_k, the orbit of z for times 0,1,...,n
stays close to the orbit of x, and for times -n,...,-1,0 the orbit of z
stays close to the orbit of y, with the closeness tending to zero
uniformly as k -> infinity. So z "follows x into the future and follows
y into the past."
(b) VOLUME CONTROL: The pushforward measure
(z)_* (rho_k restricted to G_k, WITHOUT renormalizing) satisfies
(z)_* (rho_k | G_k) <= 2 nu.
Condition (b) is the important one. It says: even though pairs are picked
(x,y) in cells and applying a map, the image measure is comparable to the
original nu, with factor 2. No concentration.
Contrast with the trivial attempt "z := x" for the future and "z := y" for
the past; that would not be a single point. The math is to construct a
single point z whose orbit shadows x in the future and shadows y in the
past. And the volume control condition says "the solution set of that
boundary-value problem does not shrink significantly."
WHY DOES THIS EXIST AT ALL FOR A SMOOTH VOLUME-PRESERVING MAP? Not because S
is hyperbolic. Pesin theory is never used. S may have zero Lyapunov
exponents everywhere, or all its exponents may vanish. The construction uses
only finite-time linear algebra on one good dyadic segment at a time.
On such a segment, for each |t| <= n the derivative DS_t written in
orthonormal frames that move with the orbit (QR / Gram-Schmidt) is upper
triangular with positive diagonal e^{lambda_1},...,e^{lambda_d}. Order these
rates from largest to smallest, and pick a threshold gamma that falls in a
gap of size at least g = b_k^{1/2}. The gap exists because the chord lemma
pins each diagonal log-sum to within b_k*n (tiny vs n), and b_k -> 0 slowly
but to zero.
Call the top p rates FAST (expanded forward) and the bottom d-p rates SLOW
(expanded backward). On any subinterval of length l: the fast block grows
forward at rate about e^{Lambda*l}, Lambda = sum of fast rates; the slow
block grows backward at rate about e^{Omega*l}, Omega = minus sum of slow
rates; every mixed p-form (a p-form with at least one slow factor) grows
forward at rate at most e^{(Lambda-2g)*l}. That 2g gap is the whole engine:
the fast leading minor dominates every mixed minor by a factor e^{2g*l}, so
the Jacobian cleanly becomes 1 + o(1) even when the rates are tiny or
vanish. If the rates cluster, one group may be empty (p = 0 or p = d); the
estimates still hold with empty-group conventions.
Given x and y in the same small cell (so x and y are very close!), the
switch operates as follows. The point z is sought so that its forward fast
component matches x's forward fast component, and whose backward "slow"
component matches y's backward slow component. The freedom left over (one
component at each end) is used to solve the linear equation. A perturbation
then gives a nonlinear solution by a fixed-point argument.
WHY THE JACOBIAN COMES OUT 1 + o(1). There are two "obvious" reasons the
Jacobian might be bad:
- the expanding directions at time n could have huge or tiny volume ratio;
- the contracting directions at time -n could have huge or tiny volume ratio.
But TWO complementary switches are used. z uses the future of x and the past
of y. Also define w using the future of y and the past of x. Then in the
pair map (x, y) |-> (z, w) the large volume ratios cancel EXACTLY. The full
derivative of the pair map has determinant 1 + o(1). Since the map is a PAIR
of points, the Jacobian 1+o(1) is exact to leading order.
WHY THE INJECTIVITY MATTERS. The switch has to be a well-defined map, not a
correspondence. Two things are needed:
- the boundary-value problem has a UNIQUE solution in a large enough
neighbourhood (contraction mapping, uniqueness ball), and
- different inputs (x,y) do not lead to the same output (z,w).
The proof establishes both. The uniqueness radius is bigger than the
shadowing radius, and there is a positive wall margin in the cells so
outputs come back into the same cell, and their endpoint bins are recovered
from their endpoint values.
WHAT COULD NOT BE DONE WITHOUT SMOOTH POSITIVE DENSITY. The injectivity
argument uses the density: it needs f > 0 everywhere so that "the output
density at z is at most 2 times the input density at (x,y)". A measure that
is only singular or only absolutely continuous but not smooth would let the
pair map concentrate.
The switch needs, for each dyadic n = 2^k, a small exceptional set of points
where the derivative control of the orbit fails. The proof shows that the
failure set shrinks fast enough that summing over k converges.
The key technical tool is a stationary estimate.
A STATIONARY PROCESS. A sequence of real random variables (Y_t)_{t in Z} is
stationary if translating the indices does not change the joint law of any
finite list:
(Y_{t_1 + s}, ..., Y_{t_m + s}) has the same law as (Y_{t_1}, ..., Y_{t_m}).
So it is invariant in law under the shift map. Examples: i.i.d. sequences,
periodic sequences with random phase, orbit of a smooth function along a
measure-preserving T.
BLOCK AVERAGES AND CHORDS. For h >= 0, let
A_h = (1/2^h) sum_{t=0}^{2^h - 1} Y_t
be the average on the first 2^h terms, and let Z_h be
Z_h = (1/2^h) max_{0 <= u <= 2^h} | sum_{t=0}^{u-1} Y_t - u A_h |.
Z_h measures how much the cumulative sum drifts away from the straight chord
of slope A_h. This is exactly the deviation of a random walk from its chord,
at scale 2^h.
LEMMA (dyadic chord estimate). For a real stationary process with finite
second moment, define
v_l = E(A_{l-1}^2) - E(A_l^2), l >= 1.
Then v_l >= 0, and sum_{l >= 1} v_l <= E(Y_0^2), and for h >= 1,
||Z_h||_{L^2} <= (1/2) sum_{a=0}^{h-1} 2^{-a/2} sqrt(v_{h-a}).
Consequence: sum_{h >= 1} ||Z_h||_{L^2}^2 <= C E(Y_0^2) for an absolute
constant C.
IMPORTANT: this is a SUM OF SQUARES being finite, not just a sup. That is
what gives summable dyadic control: at scale k, the L^2 error in the "chord
deviation" is bounded by a quantity whose SQUARED sum is finite.
INTERPRETATION. Think of the partial sums S_u = Y_0 + ... + Y_{u-1} as a
random walk with drift A_h at scale 2^h. The lemma says that the maximum
over ALL intermediate times u in [0, 2^h] of |S_u - u*A_h| / 2^h has L^2
norm whose squares sum to something finite over dyadic h. That is a strong
statement: it is the smooth-dynamics equivalent of "a martingale has a
quadratic-variation-driven central limit", but in L^2 rather than
distributional form.
THE APPLICATION: TRIANGULAR COCYCLE. On a d-manifold with an S-invariant
probability, consider the orthonormal FRAME BUNDLE F M. That is: over each
point x is a fiber of orthonormal identifications e : R^d -> T_x M. This is
a finite-dimensional bundle. For any frame e over x, form the
positive-diagonal QR factorization
D S_x o e = tilde S(e) o R(e),
where tilde S(e) is a new frame over S x and R(e) is upper triangular with
positive diagonal. This is ordinary Gram-Schmidt: R is the triangular shape
of derivative, tilde S(e) is the orthonormalized result. The map e |-> tilde
S(e) is a homeomorphism (indeed C^infinity) of the frame bundle.
Then, choosing a tilde S-invariant probability tilde nu on the bundle that
projects down to the original nu, the quantities g_j(e) = ln(R(e)_{jj}) are
bounded continuous functions (compactness of F M gives uniform boundedness
of R(e) and R(e)^{-1}) and
(g_j(tilde S^t e))_{t in Z}
is a real stationary process.
Applying the dyadic chord lemma to each g_j and summing gives, for each
dyadic scale k and each tolerance eta, a compact "good set" of base points
in M (call it G_k) with nu(G_k) >= 1 - eta/100, such that on G_k the
cumulative log-diagonals of the triangular cocycle deviate from their chord
at most b_k n, where b_k is the small positive sequence b_k >= 2^{-k/10}
with sum_k b_k^2 < infinity. This is the same b_k whose eighth root B_k
controls the cell count.
This gives the "good finite orbit segments" property: within the good set,
derivative growth along the orbit is uniform on the fast directions and on
the slow directions, at every subinterval of a good segment, with a small
additive error that is summable over dyadic scales.
WHY THE SUMMABLE-SQUARED RATE MATTERS LATER. Thematic reason: at each dyadic
scale k an exceptional set of mass O(b_k^2). Since sum_k b_k^2 < infinity,
the "good points at all large enough scales" still have full measure. If the
rate were not summable, the "good at all scales" set could have zero
measure. This is exactly what a pointwise ergodic theorem does NOT give: an
ergodic theorem gives convergence almost surely along a sequence, but the
RATE of convergence can be arbitrarily slow. The dyadic chord lemma gives an
L^2 rate.
Given a good dyadic segment, the diagonal triangles of the derivative are
close to a chord. More is needed: the d-dimensional tangent space must be
split tangent space into a "fast" part (expanded by the future) and a "slow"
part (expanded by the past), with two properties:
- the split varies continuously with the point on the segment, and
- the split has controlled angle (the projections between the two parts
are bounded in norm).
The construction of this split proceeds in three steps.
STEP 1: SPLIT THE DIAGONAL RATES. The diagonal entries of the triangular
cocycle are e^{lambda_j} at each time step. Order them so that lambda_{(1)}
>= lambda_{(2)} >= ... >= lambda_{(d)}. Choose any real threshold gamma such
that no lambda_j is within distance g of gamma (g > 0 separation). Let F be
the set of indices j with lambda_j >= gamma + g ("fast"). Let p = |F|, so p
is the dimension of the fast part.
Define:
Lambda = sum_{j in F} lambda_j (fast rate sum)
Omega = - sum_{j not in F} lambda_j (slow rate sum, note negative)
STEP 2: CONJUGATE TO HAVE INVARIANT BLOCKS. The triangular cocycles are
triangular, but not block-diagonal. A conjugating change of basis sets B_t =
C_{t+1}^{-1} A_t C_t where C_t are upper-triangular changes of basis, chosen
so that A_t becomes block-diagonal with the two groups F and complement F.
This is done by processing super-diagonals in increasing order and using
elementary "shear" matrices Q_t = Id + u_t E_{ij} at each off-block
position. Each shear is solved via a one-step linear recurrence u_{t+1} = q
u_t + f_t with |q| < 1 or |q| > 1 (the separation of rates guarantees |q| !=
1). The C_t are uniformly bounded by e^{C B n}, since the shear bounds are
polynomial in n and ln(n) <= B n.
STEP 3: ESTIMATE EXTERIOR POWERS. After conjugation, the derivative splits
into a fast block (upper triangular, dimension p) and a slow block (upper
triangular, dimension d-p). Then:
- the fast block's p-form growth rate is Lambda l + O(B n);
- the slow block's (d-p)-form growth rate is Omega l + O(B n);
- mixed forms with at least one slow vector in a p-form product have growth
rate (Lambda - 2g) l + O(B n);
- and similarly backward with Omega - 2g.
The gap "2g" is the KEY. The mixed forms are strictly slower by 2g than the
pure fast p-form. This makes the leading p-form DOMINATE: the leading minor
is roughly e^{Lambda n}, every mixed minor is roughly e^{(Lambda-2g)n}, so
their ratio is e^{2g n} and the leading term is exponentially bigger than
every other. In the Jacobian computation the leading term therefore
determines the sign and magnitude of the full determinant, and the other
minors are relatively negligible.
The 2g is the separation margin between "fast" and "slow" rates. It is what
allows the split to be meaningful and to produce a well-conditioned
Jacobian.
STEP 4: COMMON SPACES IN A SPATIAL BIN. At scale k, M is partitioned into
SPATIAL BINS: small balls (or boxes in charts) of radius roughly h_0 = e^{-B
n}. For each bin U, a SINGLE threshold gamma_U and a SINGLE pair of
reference spaces E_U^s, E_U^f that work for ALL good points in U.
This works because:
- all good points in U have their sorted rate lists agreeing to within
o(g) (via the chord control and the short derivative comparison);
- there are only d+1 candidate thresholds (1+8j)g for j = 0,...,d, and at
most one of them is within 3g of a given rate;
- so there is a candidate that separates every list with margin > 2g.
Given a threshold, the reference spaces are chosen once per bin (via a fixed
good witness in the bin). All actual spaces in the bin are within e^{-c g L}
of the reference, where L is a short dyadic length (roughly n b^{1/4},
larger than the dominant short error). This is because of a "short map
comparison" argument: at scales short compared to n, the derivative of S^L
at nearby points has nearly identical singular vectors, with error decaying
like e^{-Bn}.
STEP 5: RELATIVE STABILITY UNDER PERTURBATION. The Jacobian argument uses
p-form products over LONG times. These products have ASTONISHINGLY small
volumes (e^{Lambda n}, Lambda can be very negative for a contracting
direction). The required estimate: if a nearby point has slightly perturbed
derivative, its p-form product is still comparable to the unperturbed one.
This is a relative perturbation estimate:
|| bigwedge^p Psi(m,0) - bigwedge^p A(m,0) || <= e^{sigma m + a}
* ((1 + C epsilon e^{a+|sigma|})^m - 1),
where the A_t are the true matrix, the Psi_t = A_t + E_t are the perturbed
ones, epsilon is the size of the perturbation, and sigma is the volume
growth rate. This inequality has a factor e^{sigma m + a} AND a factor (1 +
c)^m - 1. The 1 + c is where the perturbation acts. So if m epsilon is small
(perturbation is very small for many steps), the "1 + c" stays close to 1.
This allows the Jacobian argument to be robust: even though the volume in a
given p-form is microscopic, its PERTURBATION is even smaller by a factor
that tends to zero.
The linear data of the previous paragraphs is available at every dyadic
scale. The task is to turn that into the actual geometric construction
claimed by the switching theorem. Five steps complete the construction.
STEP A: FIX SCALES AND THRESHOLDS. At dyadic scale k, n = 2^k. Pick the same
b_k as before (from the stationary estimate), and define:
B = b_k^{1/8} (used for cell sizes and counts)
g = b_k^{1/2} (rate separation margin)
L = ceil(b_k^{1/4} n) (short comparison time)
h_0 = e^{-B n} (coarse bin side)
r = e^{-2 B n} (product cell side, much smaller than h_0)
delta= e^{-3 B n / 2} (contraction uniqueness radius)
The scale hierarchy is:
log n < b_k n < g L < g n < L < B n,
with each "<" meaning o( ), and the gap growing.
Also fix a tolerance eta (like 0.01). And a small number tau (like 0.001)
for wall margins inside cells. Neither depends on k.
STEP B: THREE PARTITIONS. At scale k three partitions of M are used.
1. COARSE BINS. Cover M with finitely many coordinate charts, each with a
smaller "inner box". Inside each inner box, grid into coordinate cubes
of side h_0. These are the coarse bins. Their total number is at most
C h_0^{-d} = exp(C d B n) = exp(O(B n)).
2. PRODUCT CELLS. For each coarse bin Q that meets the good set at scale k,
a plain threshold gamma_Q and reference spaces E_Q^s, E_Q^f are chosen. In the coordinate chart of Q, tile by
parallelepipeds spanned by these two spaces with side length r. These
are the product cells. The forward expansion of the change-of-basis map
is at most sqrt(2) (E_Q has orthonormal columns), so diameter is at most
C_d r. The inverse norm is at most e^{C B n}, so the volume of one
product cell is at least e^{-C B n} r^d up to constants. Total number of
product cells is at most exp(C B n) r^{-d} = exp(C' B n).
3. ADJUSTED CELLS. Add one extra cell for all omitted points and bins not
meeting the good set. This is the partition D_k.
EFFECTIVE PARTITION SIZE. #D_k <= exp(C B_k n).
STEP C: WHICH PAIRS SURVIVE. A pair (x, y) in M x M is "retained" if it lies
in the square of one product cell Q' x Q' and in addition:
- both x and y lie in the good set at scale k;
- both x and y are in the interior of their product cell, at distance at
least tau r from each of the product-coordinate faces;
- S^n x, S^{-n} x, S^n y, S^{-n} y each lie at distance at least tau h_0
from the faces of their coarse endpoint bins;
- if M has boundary, no iterate at integer time in [-n, n] is within
c_0 h_0 of the boundary, for a fixed c_0.
The volume of points excluded by wall strips is at most C_d tau of nu (times
fixed constants), and the volume of points excluded by any other condition
is at most eta/100 of nu, uniformly. So rho_k(G_k) >= 1 - eta for a fixed
eta: removing wall strips and neighborhoods of the boundary and bad points
costs at most eta in mass.
STEP D: THE BOUNDARY-VALUE PROBLEM. For a retained pair (x, y):
- Define four endpoint projections:
F_x(u) = P^+_{Q,U_x} o chi_{U_x} o S^n(u)
F_y(u) = P^+_{Q,U_y} o chi_{U_y} o S^n(u)
P_x(u) = P^-_{Q,V_x} o chi_{V_x} o S^{-n}(u)
P_y(u) = P^-_{Q,V_y} o chi_{V_y} o S^{-n}(u)
chi_U is the fixed localization map in the coarse bin U; P^+ is the
orthogonal projection onto the fast space of the reference endpoint bin;
P^- is the orthogonal projection onto the slow space of the past endpoint
bin.
- Seek z and w satisfying:
F_x(z) = F_x(x), P_y(z) = P_y(y)
F_y(w) = F_y(y), P_x(w) = P_x(x).
That is: z's future "arrives where x's future arrives" (from the fast
perspective) and z's past "departs from where y's past departs" (from the
slow perspective). Similarly for w with x and y exchanged.
This is eight equations in the joint unknowns z and w. But z alone solves a
closed system (first line), and w alone solves a closed system (second line).
Each is 2n time steps long. The linearized version (around the reference
orbits of x and y) has a matrix form.
STEP E: THE LINEAR INVERSE. Write the linearized system in "weighted"
coordinates. On the future half, put v_tilde_t = e^{-gamma t} v_t. This
strips out the average growth. Then:
- on the slow direction (expanded by past iterations of S), the weighted
future propagator has norm at most e^{C B n} e^{-g |t|}. It decays as
time increases forward;
- on the fast direction (expanded by future iterations of S), the weighted
past propagator decays as time increases backward.
So the future side propagates "well" in one direction, the past side in the
other. Composing, all forced terms are bounded by e^{C B n} times the
sup-norm of the data, and the two "kernel graph" corrections have norm at
most e^{-2 g n}.
Then the map (matching at time 0) is an isomorphism from E_0^s(x) oplus
E_0^f(y) onto the tangent space at the starting point. The inverse has
unweighted norm at most Q_k = e^{C g n}. That is the crucial bound: Q_k is
exponential in g n but g n is much smaller than B n (by the scale
hierarchy), so Q_k r and Q_k delta both tend to zero. This is what makes the
contraction argument work.
STEP F: NONLINEAR SOLUTION BY CONTRACTION. The nonlinear problem is v =
L^{-1}(a_0 + R(v)), where a_0 is the inhomogeneous data of size O(r) and
R(v) is the quadratic Taylor remainder of the step maps. Because all
two-variable second derivative bounds are uniform (compactness of M and its
charts), there is a fixed constant C with
|R(v)| <= C |v|^2 and
|R(v) - R(u)| <= C delta |v - u| for |v|,|u| <= delta.
The map v |-> L^{-1}(a_0 + R(v)) sends the delta-ball into the delta/2 ball
and is a contraction. So there is a unique fixed point v* in the delta-ball,
with |v*| <= 2 Q_k r. Its base point is z. Because Q_k r << delta and delta
<< h_0, the actual orbit of z stays in M and in a fixed neighbourhood of the
reference orbits. The shadows are close to x in the future and y in the
past, with error at most e^{C g n} r, which tends to 0.
In product coordinates, z differs from the "ideal exchange" z_0 = (y_s, x_f)
by at most o(r). Same for w. And z, w both stay in the original product
cell, at distance at least 3 tau r / 4 from each face, and their endpoints
stay in the four original coarse endpoint bins.
STEP G: THE JACOBIAN. Now the key: compute the Jacobian of the full pair map
(x, y) |-> (z, w).
Define H_{ab}(u) := (F_a(u), P_b(u)) for a, b in {x, y}. This is a
d-dimensional map from the manifold to a product of two Euclidean spaces of
dimensions p_Q and d - p_Q.
Claim: the four derivatives DH_{xx}(x), DH_{yy}(y), DH_{xy}(z), DH_{yx}(w)
are all invertible, and their determinants satisfy (up to 1 + o(1)):
|det DH_{xx}(x)| = A_x B_x
|det DH_{yy}(y)| = A_y B_y
|det DH_{xy}(z)| = A_x B_y
|det DH_{yx}(w)| = A_y B_x
where A_x = leading p-form minor of DS^n at x, B_x = leading (d-p)-form
minor of DS^{-n} at x, etc. The cross-terms cancel because the endpoint
projections were chosen to isolate one form at each end.
Then the exact identity:
|det D(z,w)(x,y)| = |det DH_{xx}(x)| |det DH_{yy}(y)|
/ |det DH_{xy}(z)| |det DH_{yx}(w)|,
which is exactly the ratio 1 + o(1): A_x B_x A_y B_y / (A_x B_y A_y B_x) =
1.
The relative perturbation lemma shows that the
perturbations of the four minors do not destroy this: |det DH_{xy}(z) - A_x
B_y| / (A_x B_y) -> 0, similarly for the others. So the Jacobian is
genuinely 1 + o(1), not just "approximately one in average".
STEP H: PUTTING IT TOGETHER: MEASURE INEQUALITY. Take the countable
measurable partition of G_k into pieces where the cell and the four endpoint
buckets are fixed. On each piece, (x, y) |-> (z, w) is a smooth local
diffeomorphism with Jacobian 1 + o(1).
Apply the change-of-variables formula, and sum over pieces. Because
injectivity holds (see below), the images are disjoint. And the product of
two smooth positive densities has bounded ratio to any nearby density, so
summing gives:
(z_k)_* (rho_k on G_k) <= 2 nu.
This is (b) of the switch theorem.
STEP I: INJECTIVITY. Suppose (z, w) has two preimages (x, y) and (x', y').
First, cell Q must be the same because both outputs are in the interior of
their common product cell and cells have disjoint interiors. That recovers Q
and the choice of reference spaces E_Q^s, E_Q^f.
Then the four endpoint bins U_x, V_x, U_y, V_y are recovered from the actual
endpoint values: bin(S^n z), bin(S^{-n} w), bin(S^n w), bin(S^{-n} z).
Positive wall margins ensure this recovery is stable.
So the competitor (x', y') must have the same cell, same reference spaces,
and same four endpoint projections.
Then x and x' both lie near z's future and near w's past; y and y' both lie
near w's future and near z's past. So all four points are close. But all
four must satisfy the same boundary equations, so a direct application of
the uniqueness-in-delta-ball argument (from STEP F) with matching datum 0
and terminal data 0 forces x = x' and y = y'.
The uniqueness uses the delta-ball, which is much bigger than the shadowing
radius, so this is robust.
STEP J: MEASURABILITY. The retained set G_k is Borel. The switch (x, y) |->
(z, w) is measurable because it is constructed as the fixed point of a
continuous map parametrized measurably by (x, y) (on countably many Borel
"locally uniform" pieces). The Jacobian identity gives the "restriction of a
smooth local diffeomorphism" property on each piece.
Next a symbolic (finite-alphabet) system is built that is going to be the
zero-entropy counterexample. It will have two crucial properties:
- its Kolmogorov-Sinai entropy is zero;
- short lists of its legal "words" have very large Shannon entropy.
And it will have a specific algebraic relation between words at past and
future windows, which the smooth switch cannot satisfy.
THE ALPHABET: NO; VECTOR LABELS ARE REQUIRED. Let s be a positive integer. Let F_2 = {0, 1} with addition mod 2. Let
F_2^{2s} be the (2s)-dimensional vector space over F_2. On F_2^{2s} put a
nondegenerate ALTERNATING FORM (a "symplectic pairing"):
[u, v]_s = sum_{a=1}^{s} ( u_a v_{s+a} + u_{s+a} v_a ) in F_2.
This is a binary-valued function: it takes each pair of vectors to 0 or 1.
It is linear in each argument. It is alternating: [u, u]_s = 0 always.
Nondegeneracy: if [u, v]_s = 0 for every v then u = 0.
Pick s with K >= 256 where K = floor(sqrt(ell)) for some ell defined below.
Take the alphabet A := A_1 := F_2^{2s_1} \ {0}, the nonzero vectors of
F_2^{2s_1}.
Each letter of A is a 2s_1-bit string. And a word is a finite sequence of
such strings.
THE HIERARCHY: LEVELS OF WORDS. A sequence of alphabets and word families is
built:
level 1: alphabet A_1 (nonzero vectors of F_2^{2s_1}), each letter has length
2s_1 bits. There are L_1 = 2^{2s_1} - 1 letters. Each is called a
level-1 word. m_1 = 1 (level 1 words are one letter long).
level 2: a family W_2 of L_2 words, each made of N_1 letters from A_1. Each
level-2 word has m_2 = m_1 N_1 = N_1 symbols.
level 3: a family W_3 of L_3 words, each made of N_2 words from W_2. Each
level-3 word has m_3 = m_2 N_2 = N_1 N_2 symbols.
...
level i: a family W_i of L_i words, each made of N_{i-1} words from W_{i-1}.
Each level-i word has m_i = m_1 N_1 N_2 ... N_{i-1} symbols.
The scaling parameters are chosen so that L_i and m_i grow in controlled
ways:
eps_i = 2^{-i-2}
L_i = 2^{2s_i} - 1
ell_i = log L_i
K_i = floor(sqrt(ell_i))
N_i = 2^{K_i}
H_i = 2^{floor(K_i/2)}
D_i^* = 2^{floor(K_i/8)}
m_1 = 1, m_{i+1} = m_i N_i
and the (s_i) are recursive: 2s_{i+1} is the greatest even integer at most
X_i := (eps_i / 100) * N_i * ell_i.
K_1 >= 256 is enough. Nothing depends on the exact value of s_1 beyond that.
Once level i is built, its L_i words are identified with the nonzero vectors
of F_2^{2s_i} via an arbitrary fixed bijection, so each level-i word u
carries a label vector, allowing the test [u, v]_i for u, v in W_i.
For a word w in W_{i+1}, write c_i(w) for its list of constituent level-i
words: c_i(w) = (u_0, u_1, ..., u_{N_i-1}) with each u_j in W_i. Unambiguous
because all level-i words have length m_i.
Note: later, N_i plays the role of "how many constituents in a level-(i+1)
word"; H_i is the "group size" used in the graph condition; D_i^* bounds the
window lengths and prescribed power-of-two lags, all MUCH smaller than N_i.
THE FOUR PROPERTIES. Each family W_i of L_i words, each of length m_i,
satisfies:
(W1) GRAPH RELATIONS WITHIN GROUPS.
For every w in W_{i+1}, with
c_i(w) = (u_0, ..., u_{N_i-1}), and for every 0 <= b < N_i / H_i,
every 0 <= k <= floor(K_i / 8), and every 0 <= r < r + 2^k < H_i,
[ u_{b H_i + r}, u_{b H_i + r + 2^k} ]_i = 0.
That is: within each "group" of H_i consecutive constituents, positions
separated by a power of two must be orthogonal (with respect to the
symplectic form at level i).
(W2) DISTANCE BETWEEN WORDS AND SHIFTED WORDS.
If v, w in W_{i+1} are distinct, then
d_H(c_i(v), c_i(w)) >= (1 - eps_i) N_i
where d_H counts how many constituent positions differ.
More strongly: for all v, w, z in W_{i+1} (repetitions allowed) and every
offset 1 <= t < N_i,
d_H( (c_i(v) c_i(w)) [t, t + N_i), c_i(z) ) >= (1 - eps_i) N_i.
The string (c_i(v) c_i(w)) is the concatenation of the two constituent
lists,
and [t, t+N_i) cuts a shifted window of length N_i. So any window crossing
the boundary between two concatenated words looks very different from any
legal word. That is comma-free-ness, a synchronization condition formalized in coding
theory by Tonchev (2003). ./tonchev-2003-difference-systems-sets.pdf
(W3) DISTRIBUTIONS OF SHORT WINDOWS.
Let U be uniform on W_{i+1}. For each 0 <= t <= N_i - D_i^*, there is a
probability Q_{i,t} on W_i^{D_i^*} such that
TV( law of c_i(U)[t, t + D_i^*), Q_{i,t} ) <= 10^{-4}.
Moreover the Q_{i,t} satisfy an atom bound: for 1 <= d' <= D_i^*,
distinct positions 0 <= r_1 < ... < r_{d'} < D_i^*, and any prescribed
labels v_1, ..., v_{d'} in W_i,
Q_{i,t}{ u : u_{r_j} = v_j for all j } <= 2 * (2^{2 K_i} / L_i)^{d'}.
The right side is a small number -- exponentially small in d' * K_i.
(W4) APPROXIMATION IN EACH INDIVIDUAL WORD.
For every fixed w in W_{i+1}, write c_i(w) = (u_0, ..., u_{N_i-1}) and
take J uniform on {0, ..., N_i-1}. Suppose U is uniform on W_i. Then for
each 0 <= t <= N_{i-1} - D_{i-1}^*,
TV( law of c_{i-1}(u_J)[t, t + D_{i-1}^*), law of c_{i-1}(U)[t, t +
D_{i-1}^*) ) <= 10^{-4}.
That is: short internal windows of a fixed parent word look statistically
like short windows of a uniform lower-level word, up to 10^{-4} in
total variation. This is a per-word condition, not an average over all
of W_{i+1}. This is what makes the eventual invariant measure "see"
near-uniform short windows even when it weights some parents more than
others.
Total variation distance: (1/2) * sum over a of |p(a) - q(a)|.
PROOF OF COMPATIBILITY. Proposition 10.1 (Finite word families,
restatement). For every s_1 with K_1
>= 256, the recursion above produces families W_i of L_i words, each of
length m_i, satisfying (W1)-(W4) at every level.
Proof, in four stages.
Stage 1: SAMPLE LISTS WITH THE GRAPH RELATIONS. Fix W_i and its vector
labeling. Generate the constituent list (U_0, ..., U_{N_i-1}) of a W_{i+1}
word in independent groups of H_i consecutive positions. Within each group,
process from left to right; choose each new label UNIFORMLY among the
nonzero vectors orthogonal to all earlier labels at the specified
power-of-two lags within the same group. There are at most K_i linear
constraints at any point, so the kernel has codimension at most K_i, so the
number of allowed labels is at least
2^{2s_i} / 2^{K_i} (roughly),
which is a big number since 2s_i is much bigger than K_i.
Call this sampling law P_i^0. Every list it produces satisfies (W1). Also
P_i^0 gives each individual position a UNIFORM marginal on W_i: this is
because the symplectic group Sp(2s_i, F_2) acts transitively on nonzero
vectors, and the sampling rule is equivariant under that action.
Stage 2: ENFORCE (W4) BY CONDITIONING. For each parent word, look at the
empirical distribution of short windows from the constituent labels. If it
is too far from the uniform law, toss the parent out. A lemma says: if Y_1,
..., Y_q are independent random probability vectors on a set of size A, all
with the same mean p, then
E TV( (1/q) sum Y_b, p ) <= (1/2) sqrt(A/q).
This Chebyshev-like bound applies to the q_i = N_i / H_i independent groups.
Markov + union over the N_{i-1} offsets gives: the probability that some
parent word violates (W4) is less than a quantity F_i, verified below is
small.
Let E_i be the event that all (W4) requirements hold. Then P_i^0(E_i) >= 1/2
(once F_i < 1/2 is verified). Define P_i := P_i^0( . | E_i ).
Stage 3: ENFORCE (W3) AND (W2) BY FRESH SAMPLING. Sample M_i := L_{i+1}
independent lists with law P_i. For each offset, the empirical law of short
windows over the samples estimates Q_{i,t}. Another application of the
averaging lemma + Markov + union over N_i offsets shows that (W3) fails with
probability < 1/4.
For (W2): reference law = each distinct list is uniform on W_i^{N_i}. A
"distance failure" means two lists agree at more than eps_i N_i positions,
or a shifted window agrees with a legal word at more than eps_i N_i
positions. For any fixed set of matching positions, form a multigraph whose
vertices are (list-index, position) pairs and whose edges are the asserted
equalities. Each connected component with v vertices imposes at least v-1
equalities: probability at most L_i^{-(v-1)}. So a set of e matching
positions has probability at most L_i^{-e/2}. Summing over all possible
match sets and comparing to P_i via a likelihood ratio gives a union bound
T_i, shown small below.
Stage 4: COMBINE. The union bound shows: at each level, the probability that
a randomly sampled family fails ANY of (W4), (W3), (W2) is < 1. So the
family exists with all properties. Detail: at most (1/2) + (1/4) + (1/4) = 1
of mass can fail, so some family works.
Finally: because (W2) with v != w forces distance N_i, the sampled words are
distinct (else distance would be 0), and there are exactly L_{i+1} of them.
That is the family W_{i+1}.
The bounds F_i, E_i, T_i are verified to be < 1/2, 1/4, 1/4 at every level
via explicit numerical inequalities just below. The estimates hold because
K_i grows extremely fast (K_{i+1} >= 2^{K_i/3} >= K_i^2 >= 4 K_i).
CONSEQUENCE 1: RECOGNITION OF WORDS AND ALIGNMENT. Lemma 10.2 (Recognition).
Define
d_i = prod_{j=1}^{i-1} (1 - eps_j),
d_* = prod_{j=1}^{infinity} (1 - eps_j) > 0.
Note d_* >= 1 - sum_j eps_j = 3/4.
(a) Distinct v, w in W_i satisfy d_H(v, w) >= d_i m_i >= d_* m_i.
(b) For all v, w, z in W_i and every offset 1 <= t < m_i,
d_H((vw)[t, t + m_i), z) >= d_i m_i >= d_* m_i.
Proof. The finite products d_i are decreasing and converge to 1 - sum eps_j
= 3/4 > 0. The bound on d_H(v,w) follows by induction from (W2) applied to
successive levels: at level i+1, two distinct words differ in at least (1 -
eps_i) N_i constituent positions; each difference contributes at least d_i
m_i symbol mismatches at level i; that gives d_{i+1} m_{i+1}.
The offset case: if m_i divides t, the constituent offset t / m_i is
nonzero, and (W2) applies directly. If not, each aligned target constituent
faces a length-m_i source window at a nonzero offset modulo m_i. That source
window crosses exactly two consecutive legal level-i words, and the
induction hypothesis (part b at level i) gives at least d_i m_i mismatches
per target constituent.
This is the "comma-free index" property: a legal word, shifted by any
nonzero amount and viewed against another legal word, looks different by at
least a fixed fraction of its length. This is what lets a smooth shadowing
orbit recover word labels.
CONSEQUENCE 2: THE RATES R_i DECAY, AND A 16-TH POWER DIVERGES.
Lemma 10.3 (Rates). Define R_i = ell_i / m_i (entropy per symbol at level
i). Then:
(a) m_i is a power of two, log m_i = sum_{j<i} K_j. (b) R_{i+1} <= (eps_i /
100) R_i, so R_i -> 0. (c) R_i >= R_1 * 200^{-(i-1)} * 2^{-(i-1)(i+20)/2},
so R_i decays at most
exponentially times a quadratic-in-i exponential correction.
(d) sum_{i >= 1} K_i R_i^{16} = infinity.
The last statement is the crucial "divergence": K_i R_i^{16} does not decay.
Physically: each level offers K_i "test scales" (dyadic positions inside a
word), and each scale carries entropy R_i per symbol. Sum over all scales is
infinite in the 16th-power sense.
Combining (b) with the growth K_{i+1} >= 2^{K_i/3} >= K_i^2: for large i,
log(K_i R_i^{16}) >= 2^{i-1} log K_1 + 16 log R_1 - 8(i-1)(i+20), which
tends to +infinity.
The comparison between smooth and symbolic is now set up: on the smooth side
the sum of B_k^{16} is FINITE, and on the symbolic side sum K_i R_i^{16} is
INFINITE. This "summable vs. non-summable" is where the eventual
contradiction will live.
The word families are the ingredients. The symbolic system glues them into a
single ergodic mpt with two crucial properties:
(a) Kolmogorov-Sinai entropy zero;
(b) short window entropy almost maximal.
That is: per-symbol entropy is tiny, but a short window of D consecutive
level-i words carries almost D * ell_i bits of entropy. And (a) and (b) can
both be true because "almost maximal" is offset by "per level i the maximum
per symbol is R_i which tends to zero".
CONSTRUCTION: THE COMPACT PHASE EXTENSION. Let A = W_1 be the level-1
alphabet (nonzero vectors of F_2^{2s_1}). Take the compact space
Omega = A^Z x prod_{i >= 1} (Z / m_i Z).
A point is (omega, a) with omega: Z -> A a bi-infinite sequence of letters
and a = (a_1, a_2, ...) with a_i in Z/m_i Z. The coordinate a_i has a
MEANING: it is the position of the origin relative to the boundary of the
level-i word containing it.
Define a subspace X of Omega by the conditions:
(i) COMPATIBILITY. a_{i+1} = a_i mod m_i for every i. So higher phases
refine lower phases.
(ii) LOCAL LEGALITY. Whenever t = -a_i mod m_i (i.e. t is a level-i word
boundary), the length-m_i block
(omega_t, omega_{t+1}, ..., omega_{t+m_i-1}) is a legal level-i word,
i.e. belongs to W_i.
X is closed in Omega (finite constraints on each coordinate from a closed
set), so X is compact. On X the shift
T(omega, a) = (omega', a'), omega'_t = omega_{t+1}, a'_i = a_i + 1 mod m_i
is a homeomorphism.
NONEMPTY. For any j, take a word from W_j and align it bi-infinitely with
boundary at 0. Since higher-level words are concatenations of lower-level
ones, this sequence satisfies all conditions through level j. These finite-
level sets are nested; compactness in X gives an infinite-point
intersection, which is nonempty. Alternatively: by Konig's lemma there is an
infinite path through the tree of consistent finite sequences.
INVARIANT MEASURES EXIST by compactness. A self-contained treatment of the
invariant-measure and ergodic-theoretic background is in Dajani and Dirksin
(2008). Weak limits of empirical averages (1/n) sum_{j=0}^{n-1} delta_{T^j
x} are T-invariant, so the compact convex set M_T(X) of T-invariant
probabilities is nonempty. Standard ergodic theory gives: this convex set
has extreme points, and extreme points are exactly the ergodic measures.
./dajani-dirksin-2008-simple-intro-ergodic-theory.pdf
PROPOSITION 11.1 (Invariant system on X). There is a T-invariant ergodic
probability mu on X. Every T-invariant probability on X is nonatomic, is
standard modulo null sets, and has Kolmogorov-Sinai entropy zero.
Proof pieces.
Ergodicity: pick a countable uniformly dense family of continuous test
functions. Successively restrict to the compact face of invariant
probabilities where each f_j integrates to its maximum over the previous
face. The nested intersection is a single probability (extreme because it
lies on all faces; a convex combination decomposition would contradict max
at some f_j). So it is ergodic.
Nonatomicity: for any T-invariant nu, the coordinate a_i is uniform on Z/m_i
Z, because T cyclically permutes the level sets. If nu had a measurable atom
of mass c > 0, then for every i, up to null sets, it would lie in exactly
one a_i-level class, so c <= 1/m_i. Since m_i -> infinity, c = 0.
Standard: X is a compact metrizable subspace of countable product of finite
sets. Encode finite coordinates into disjoint ternary digit blocks {0, 2};
this gives a continuous injective map of X into a compact subset of [0, 1].
The image law under mu is nonatomic so its distribution function F is
continuous, and the quantile Q is a measurable inverse. This gives a
measurable isomorphism (X, mu) -> ([0, 1], Lebesgue).
Zero entropy: this is the important one. Take any partition P depending on
only FINITELY many coordinates: omega_{-r}, ..., omega_r and a_I for one
level I. The compatibility (i) determines all lower phases a_1, ..., a_{I-1}
from a_I. And a_j for any j >= I determines the boundaries of all level-j
words intersecting [-r, r].
Fix j >= I. The interval [-r, n-1+r] used in the n-step name of the
partition meets at most ceil((n + 2r)/m_j) + 1 level-j words. Each is one of
L_j words. So the number of possible n-step names is at most
m_j * L_j^{ceil((n+2r)/m_j) + 1},
which grows as m_j * exp(R_j n (1 + o(1))). The entropy rate of P is at most
R_j = (log L_j)/m_j. Taking first j -> infinity and then letting P range
over finite partitions of the phase extension gives h_mu(T) = 0. In fact
this bound holds for EVERY invariant measure, not just the chosen mu.
Finally, an arbitrary finite partition P (with A atoms) can be approximated
by a finite-coordinate partition Q with mismatch probability s arbitrarily
small, because cylinder sets generate the Borel sigma field. Chain rule:
H_n(P) <= H_n(Q) + H(P | Q),
and
H(P | Q) <= H_bin(s) + s log A.
Divide by n, take n -> infinity, then s -> 0. This shows h_mu(T, P) = 0 for
every finite P, so h_mu(T) = 0 for the phase extension.
ALIGNED WINDOWS AND THE WINDOW-ENTROPY STRUCTURE. Fix once and for all an
ergodic mu on X (from Proposition 11.1). For x = (omega, a) in X, define the
level-i word containing the origin:
V_i(x) in W_i, the string of length m_i beginning at the boundary
-a_i(x) that contains position 0.
And let A_i = {x in X : a_i(x) = 0} be the "level-i boundary at origin" set.
Define the CONDITIONAL MEASURE
mu_i := m_i * (mu restricted to A_i),
which is a probability (uniformizing the phase at origin).
The boundary-start law mu_i is the key object. It says: pick a point in X at
a level-i boundary, look at the words that begin at that boundary.
Define the advance function r_i(x, t) in {0, ..., m_i - 1}: it is ( -
a_i(T^t x) ) mod m_i, the distance from t to the next level-i boundary at or
after t.
Define, for D >= 2, the ALIGNED LIST
L_{i,D}(x, t) = ( V_i(T^{t + r_i(x,t) + j*m_i} x) )_{j = 0}^{D - 2}.
So L_{i,D}(x, t) is the list of the first D-1 full level-i words starting at
the first level-i boundary at or after t.
Lemma 11.2 (Aligned law). The law of L_{i,D}(x, 0) under mu is the same as
the law under mu_i of the first D-1 level-i words beginning at zero. Under
mu_i, the parent slot index
J := a_{i+2} / m_i
is uniform on {0, 1, ..., N_i N_{i+1} - 1}, and is independent of V_{i+2}:
for each such j and each v in W_{i+2},
mu_i(J = j, V_{i+2} = v) = mu_{i+2}(V_{i+2} = v) / (N_i N_{i+1}).
Proof. Invariance allows t = 0. Phase change: T^{-a_i} maps the level-i
phase class onto A_i, so mu_i captures the boundary-start law. Slot
uniformity: T^{j*m_i} permutes the sets {a_{i+2} = 0, V_{i+2} = v}
cyclically, so each has the same mu-mass. Multiply by m_i and use
m_{i+2}/m_i = N_i N_{i+1}.
The uniformity (ii) is what makes "every parent word conditioned" turn into
"every offset sampled uniformly" in the entropy computation.
WINDOW ENTROPY IS ALMOST MAXIMAL. Two elementary entropy bounds are needed.
The second is a standard maximal-coupling argument; see Sason (2013).
./sason-2013-entropy-bounds-maximal-coupling.pdf
Bound 1. If a law on a finite set has every atom of mass at most M, then its
entropy is at least log_2 (1/M).
Bound 2. If two laws P, Q on an A-point set satisfy TV(P, Q) <= s, then
H(P) >= H(Q) - 1 - s log_2 A.
Proof of Bound 2: couple them with mismatch probability <= s, record the
mismatch flag, chain rule.
Proposition 11.3 (Window entropy). For all sufficiently large i, for every
power of two D with 2 <= D <= D_i^*, and every integer t:
H_mu( L_{i,D}(., t) ) >= .996 * (D - 1) * ell_i.
ell_i = log_2 L_i, and the threshold on i is independent of D, t, and the
specific invariant mu on X.
Proof. Work under mu_i (the boundary-start law). Write q = D - 1. Split the
uniform parent slot J as
J = h*N_i + u, 0 <= h < N_{i+1}, 0 <= u < N_i.
Conditional on the parent v = V_{i+2} and the parent-internal offset u, the
slot h is uniform on {0, ..., N_{i+1}-1}. Call u ADMISSIBLE if u + D_i^* <=
N_i. The non-admissible fraction is beta_i := (D_i^* - 1)/N_i, tending to 0.
For fixed v and admissible u:
- By (W4) at level i+1, the law of the D_i^*-window at offset u is within
10^{-4} in TV of the law of the same window in a uniform word of W_{i+1}.
- By (W3) at level i, that law is within a further 10^{-4} of Q_{i,t}, whose
atom at any prescribed q-position tuple is at most
M_i = 2 * (2^{2 K_i} / L_i)^q.
Restriction to the first q entries cannot increase TV distance. So by Bound
2 with s = 2 * 10^{-4} and alphabet size L_i^q, the actual q-list
conditionally on (v, u) has entropy at least
q * ell_i - 2 K_i q - 1 - (1 + s q ell_i) = q * ((1 - s) ell_i - 2 K_i) - 2.
Mixing over (v, u) preserves this by concavity of entropy; the
non-admissible fraction beta_i only reduces this by a factor (1 - beta_i).
So
H_mu( L_{i,D}(., t) ) >= (1 - beta_i) * ( q((1-s) ell_i - 2 K_i) - 2 ).
Now beta_i -> 0, and 2 K_i / ell_i -> 0, and 2 / ell_i -> 0. Once each is at
most .0005, dividing by q*ell_i, with q >= 1:
(1 - .0005) * (1 - .0002 - .0005 - .0005) > .996.
So the ratio H / (q ell_i) is at least .996, proving the proposition.
Note: only two consecutive levels i and i+1 enter this argument. The TV
errors do NOT accumulate over levels; the total variation cost is exactly 2
* 10^{-4}.
THE GRAPH RELATION ACROSS TWO WINDOWS. The graph structure is the binary
symplectic graph of Seidel (1973). Two legal level-i words u, v in W_i pass
the GRAPH TEST at level i if
[u, v]_i = 0,
using the symplectic labels. Now consider a fixed point x in X and its two
aligned windows: the PAST window L_{i,D}(x, -n) and the FUTURE window
L_{i,D}(x, 0). Both are lists of D-1 words. Question: for how many j in {0,
..., D-2} does the pair (past entry j, future entry j) pass the graph test?
./seidel-1973-two-graphs-symplectic-gf2.pdf
Lemma 11.4 (Graph windows). Let 2 <= D <= D_i^* be a power of two and set n
= D * m_i. For all sufficiently large i, with mu-probability at least
1 - alpha_{i,D}, alpha_{i,D} := (2D - 2)/H_i,
ALL corresponding pairs of past and future entries pass the graph test at
level i simultaneously. In particular, sup_{D <= D_i^*} alpha_{i,D} -> 0.
Proof. Since n is a multiple of m_i, both windows advance by the same q_0 =
r_i(x, 0). Entry starts are
past: -n + q_0 + j*m_i
future: q_0 + j*m_i
for 0 <= j <= D-2. Corresponding entries are separated by exactly D level-i
positions. The first start through the last start SPAN 2D - 1 level-i
positions (including the unused middle block between the two windows).
Now, from the aligned law (Lemma 11.2), the first boundary position of the
past window is uniform mod H_i (because the parent-internal position is
uniform mod N_i, and H_i divides N_i). Since 2D-1 <= H_i (by D <= D_i^* and
H_i >> D_i^*), exactly 2D - 2 residues have a span that crosses some
H_i-group boundary. Every OTHER residue has all 2D-1 aligned positions
inside a SINGLE group.
But (W1) is exactly the statement that within each group, positions
separated by a power of two satisfy [u, v]_i = 0. Since D is a power of two,
this applies. So on every residue NOT crossing a group boundary, ALL D-1
pairs pass simultaneously.
The bad fraction is exactly (2D-2)/H_i, and 2D_i^* / H_i -> 0.
The graph lemma is what makes the symbolic side "rigid": nearly all pairs of
windows pass the graph test, forced by (W1). And the smooth switch, by
contrast, forces independence of the two windows, so the smooth side can
only pass the graph test rarely.
That is the setup. Now the contradiction.
THE OBSTRUCTION. The punchline: on the smooth side, matching past and
future lists are INDEPENDENT (conditional on the cell and one entry), so the
fraction of graph-test passes is at most about 62%. On the symbolic side,
the corresponding lists are FORCED to pass almost always by (W1).
Contradiction.
FIRST: FINDING CHEAP SCALES ON THE SYMBOLIC SIDE. Lemma 12.1 (Cheap scales).
Let (B_k)_{k>=1} be any POSITIVE sequence with sum_k B_k^{16} < infinity.
Then there exist a subsequence of levels i -> infinity and, at each such i,
a dyadic D with
2^{ceil(K_i/16)} <= D <= 2^{floor(K_i/8)},
such that, with n = m_i * D:
B_{log n} / R_i -> 0 along the subsequence,
where R_i = ell_i / m_i.
In particular, the smooth partition at scale log n has
log #D_{log n} = o( (D - 1) ell_i ),
for every fixed constant in the partition-size bound.
Proof. Set b_i' = log_2 m_i, then b_{i+1}' = b_i' + K_i. Consider the
integer sets
J_i := { b_i' + j : ceil(K_i/16) <= j <= floor(K_i/8) }.
These sets are disjoint (each covers a different range of dyadic scales),
and for large i the sets have at least K_i / 32 integers each.
Assume for contradiction that B_{log n} / R_i >= c > 0 for all sufficiently
late tests. Then
sum_k B_k^{16} >= c^{16} * sum_i |J_i| * R_i^{16}
>= c^{16} * (1/32) * sum_i K_i R_i^{16}
= infinity,
because sum_i K_i R_i^{16} = infinity (Lemma 10.3d). But sum_k B_k^{16} is
finite. Contradiction.
So there are scales where conditioning on the smooth partition cell loses at
most a vanishing fraction of the window entropy.
SECOND: THE SMOOTH MODEL, AND THE UPPER BOUND FROM INDEPENDENCE. Suppose for
a contradiction that (X, mu, T) has a smooth positive-volume model (M, nu,
S). Its dimension is positive because a nonatomic probability cannot live on
a zero-dimensional compact manifold (which is finite).
Apply the switch theorem with tolerance eta =
.01. All the smooth data, the sequence B_k, and the constant C are now
fixed.
Use cheap scales Lemma 12.1 for this sequence. So there is a subsequence of
i, D, n with n = m_i D and B_{log n} / R_i -> 0, and
log #D_{log n} = o( (D-1) ell_i ).
Identify X, via the smooth isomorphism, with M. So mu is carried to nu.
At each such test scale, let C be the cell of D_k sampled in the pair law
rho_k, and (x, y) be the retained pair.
Define two lists:
U := L_{i,D}(y, -n) (past window, aligned under y)
V := L_{i,D}(x, 0) (future window, aligned under x)
q := D - 1
And define the passing statistic
Z(x, y) := (1/q) * sum_{j=0}^{q-1} 1_{ [U_j, V_j]_i = 0 },
the FRACTION of corresponding entries that pass the level-i graph test.
For each fixed cell and each fixed j, the two entries U_j and V_j are drawn
INDEPENDENTLY: U_j from y's past law, V_j from x's future law, conditional
on (x, y) sharing cell C. Independence is EXACT, because rho_k conditioned
on C is a product of two copies of (nu|_C).
Upper bound on Z: by the window-entropy prop and independent marginals,
H(U), H(V) >= .996 * q ell_i.
Conditioning each vector on the sampled cell loses at most H(C) <= log
#D_{log n} = o(q ell_i). So conditional subadditivity:
(1/q) sum_{j=0}^{q-1} H(U_j | C) >= .99 ell_i,
(1/q) sum_{j=0}^{q-1} H(V_j | C) >= .99 ell_i.
Apply the graph-independent lemma (below) with index xi = (C, j), j uniform
independent of the cell. So conditional on the full index, U_j and V_j are
independent samples from two high-entropy laws on nonzero vectors.
LEMMA 12.2 (Graph independence). Let L = 2^{2s} - 1 (so L+1 is a power of 2),
equipped with a nondegenerate alternating form. Suppose a family of pairs
of laws (p_xi, q_xi) indexed by a probability space satisfies
E H(p_xi) >= .99 log_2 L, E H(q_xi) >= .99 log_2 L.
Then for all sufficiently large L, the average over xi of the probability
that independent samples from p_xi and q_xi pass [u, v] = 0 is at most .62.
Proof. For a law p, let A_p = {u : p(u) > L^{-.8}} (the "heavy"
points), and theta_p = p(A_p). Heavy entropy bound:
H(p) <= 1 + .8 * theta_p * log L + (1 - theta_p) * log L
= 1 + (1 - .2 theta_p) * log L.
So E theta_p <= .055 when log L >= 1000.
Matrix A_{uv} = (-1)^{[u,v]}: on the FULL vector space, its rows are
orthogonal (nondegeneracy makes v -> [u+w, v] a nonzero functional for
u != w, so the sum over v vanishes). Each row has squared norm L+1. So the
full operator norm is sqrt(L+1), and the nonzero-block has no larger norm.
Light subprobability p_l = p * 1_{A_p^c} satisfies ||p_l||_2^2 <= L^{-.8}.
So |p_l * A * q_l| <= sqrt(L+1) * L^{-.8} =: beta_L, and beta_L <= .02 for
large L.
The passing mass of the light-light pair is at most (1 + beta_L)/2.
Plus the heavy-heavy and heavy-light parts contribute at most
E theta_p + E theta_q <= .055 + .055.
Sum:
.055 + .055 + (1 + .02)/2 = .62.
The diagonal of A is included (u = v has [u, v] = 0), so no separate
exclusion of diagonal is needed.
Conclusion:
E_rho Z <= .62. (upper)
THIRD: THE LOWER BOUND FROM SHADOWING. The shadowing property of the switch
supplies the transition. The underlying hyperbolic-dynamics background,
including the canonical local-product coordinates, is in Smale (1967); a
shadowing precedent of the same kind is Chow, Lin, and Palmer (1989). Recall
from the geometric construction: the retained set G_k has
rho_k-mass >= 1 - eta, and for (x, y) in G_k the output z satisfies: for 0
<= t <= n, S^t z is close to S^t x; for -n <= t <= 0, S^t z is close to S^t
y; and the closeness tends to zero UNIFORMLY as k -> infinity.
The needed step translates geometric closeness into symbolic agreement. Use
a compact-cores argument.
Lemma 12.3 (Symbol matching). For all sufficiently large dyadic times n in
the smooth switching estimate, the retained pairs (x, y) in G_k for which
# { 0 <= t < n : f(S^t z) != f(S^t x) } < d_* n / 100
# { -n <= t < 0 : f(S^t z) != f(S^t y) } < d_* n / 100
have original rho-mass at least .96. f: M -> A is the transported symbol
at time 0.
Proof. Inner regularity of the smooth probability nu gives compact
sets K_a contained in f^{-1}(a), for a in A, whose union has mass at least
1 - kappa of nu, with kappa << d_* / 20000. Measurable atoms are replaced by
Borel representatives modulo null sets. Each core is compact, and the
distance between any two nonempty distinct cores is positive (uniform
separation). Uniform shadowing error at large scales is smaller than this
separation, so two compared iterates both in cores have the SAME symbol.
Write G = G_k and sigma = z_* (rho |_G). The switch gives sigma <= 2 nu, and
by pushing forward invariance, (S^t)_* sigma <= 2 nu for every t. The x and
y time marginals under rho are nu. So at each future comparison time, the
rho-mass within G where either x or z misses the cores is at most 3 kappa.
Same on the past half. Integral of mismatch fraction on each half <= 3 kappa.
Markov at threshold d_* / 100 bounds the failure mass on each half by
600 kappa / d_* < .03. Since rho(G) >= 1 - eta >= .99, joint success mass
>= .96. All estimates are in the ORIGINAL law, without renormalizing.
Now the punch line: on the "matching" event, the phases agree.
If a_i(z) != a_i(x), then on each of the D-1 full level-i words beginning at
x's first boundary in [0, n), the symbol string of z is a window of length m_i
cut at a NONZERO offset across two consecutive legal level-i words. By
recognition (Lemma 10.2), such a window differs from any legal word in at
least d_* m_i positions. Over D-1 disjoint intervals that is
>= d_* (D-1) m_i = d_* n * (D-1)/D >= d_* n / 2
mismatches, contradicting the < d_* n / 100 mismatch budget from the matching
event.
So
a_i(z) = a_i(x) = a_i(y). (phase recovery)
a_i(S^{-n} u) = a_i(u) because n is a multiple of m_i.
Then the two aligned entries U_j, V_j occupy, in the symbolic orbit of z,
the positions -n + q_0 + j m_i and q_0 + j m_i. Their DIFFERENCE is exactly
n = D m_i, and their offset from z's boundaries is uniform. And on the
matching event, the actual labels match those of z at those positions,
EXCEPT for at most n/100 * (1/2 + 1/2) = n / 100 positions total (combining
both halves), so at most D / 100 entries differ on each side.
Finally, apply the graph-window lemma (Lemma 11.4) for the ACTUAL symbolic
orbit of z: with mu-probability at least 1 - alpha_{i,D} (which is close to
1 for late tests), all D-1 pairs of z's window entries pass the graph test.
Push this through sigma = z_* (rho |_G) <= 2 nu: the mass of pairs with z in
the graph-exception set E_{i,D} is at most 2 alpha_{i,D}. Choose a late test
with 2 alpha_{i,D} <= .01.
On the intersection of:
- the matching event (mass >= .96) minus graph-exception (mass <= .01):
at least .95 of the pair space,
- and for those pairs, removing at most D/100 wrong entries per side,
if the only wrong entries are fewer than D/100 on each side, then
Z(x, y) = (1/(D-1)) * #{j : [U_j, V_j]_i = 0}
>= (D-1 - 2D/100) / (D - 1)
>= 1 - (2D/100) / (D - 1)
>= .96.
Z is nonnegative everywhere else, so
E_rho Z >= .95 * .96 >= .912. (lower)
FOURTH: THE CONTRADICTION. (upper) says E_rho Z <= .62. (lower) says E_rho Z
>= .912.
Both are statements about the SAME quantity E_rho Z, computed under the same
original pair law rho, at the same set of test scales.
The two inequalities are incompatible. Contradiction.
Since the only assumption was the existence of a smooth positive-volume
model, there is no such model.
To avoid confusion with related results, the following is a precise list of
what is and is not established.
CLAIMED: There exists ONE ergodic invertible mpt (X, mu, T) of a standard
nonatomic probability space with h_mu(T) = 0 such that (X, mu, T) has NO
smooth positive-volume model on ANY finite-dimensional compact C^infinity
manifold, regardless of boundary and orientability, with ANY smooth positive
density and ANY C^infinity volume-preserving diffeomorphism.
NOT CLAIMED: (1) That every finite-entropy ergodic mpt fails to have a smooth
positive-volume model. A single counterexample suffices. Plenty of
finite-entropy systems DO have smooth models: hyperbolic toral
automorphisms (polynomial diffeomorphisms of T^2 preserving Lebesgue
measure) are Bernoulli with positive entropy, and so are many Anosov
geodesic flows on negatively curved manifolds with the smooth
Liouville measure. Those isomorphism classes have smooth positive-
density models. The Lind-Thouvenot theorem gives mere homeomorphism
models, not diffeomorphism models, and is not an example of this
kind.
(2) That full-support invariant measures cannot occur as invariant measures
of smooth diffeomorphisms. Quas-Soo 2016 shows such measures exist, and
can be singular. Only the SMOOTH POSITIVE DENSITY restriction
is used.
(3) That the system fails to have a model with a merely absolutely-continuous
invariant measure. Foreman-Weiss 2022 treats measures that are
equivalent to volume but not necessarily smooth positive density.
(4) That the system fails to have a Lipschitz model, or a homeomorphism
model. Katok-Thouvenot 1997 treats Lipschitz and slow-entropy
obstructions for Z^2 actions; the obstruction does not apply to the
single-transformation problem. Lind-Thouvenot 1977 realizes every
ergodic invertible finite-entropy mpt by a Lebesgue-preserving
homeomorphism of the two-torus.
(5) That smooth systems are hard to classify. Anti-classification results
(Foreman-Weiss, Kunde, Gerber-Kunde) say the measure-isomorphism relation
is not Borel on certain families of smooth systems. That is a statement
about equivalence classes, not about individual realizations.
(6) That the zero-entropy system has any unusual properties beyond the
specific counterexample. It is a construction from word families. It
is ergodic, invertible, zero-entropy, standard nonatomic. That is all.
(7) That the numerical constants are optimal. The constants eta = .01
and K_1 = 256 are unoptimized, chosen for convenience rather than
tightness. A sharper argument would push these down. Their exact
values do not affect validity. The .62 vs .912 gap has slack for
further tightening. None of this matters for correctness: only
.62 < .912 does.
The core contradiction is a single number. The smooth side of the argument
bounds the expected graph-test passing statistic by about .62 at each scale
where the smooth partition is cheap. The symbolic side forces a lower bound
of about .912 at the same scales. Both bounds use the same pair law on the
same retained set at the same dyadic time. The cheap-scales lemma supplies
infinitely many scales where both hold, which is impossible. No smooth
positive-volume model, of any finite dimension, exists for the constructed
zero-entropy system.