Ergodic theory

Ergodic theory studies the long-time statistical behaviour of measure-preserving transformations.

For a measurable transformation $T$, a measure is invariant if the pushforward measure equals the original measure, or equivalently $\mu (T^{-1}A)=\mu (A)$ for every measurable $A$. This is the measure used in a measure-preserving system.

A Borel probability measure is invariant for a Markov kernel semigroup if every transition preserves it. For a Feller semigroup on a compact metric space, it suffices to test the displayed identity on continuous functions. Kernel Jensen inequality gives contraction on $L^p(\mu )$, and sup-norm strong continuity followed by density gives strong continuity there for $1\le p<\infty$.

For a Feller semigroup on a compact metric space, time-averaged transition laws have weakly convergent subsequences by compactness of probability measures on a compact metric space. Shifting a time interval of length $n$ by $t$ changes a bounded test-function average by at most $2t\Vert f{\Vert }_{\infty }/n$. A weak subsequential limit is therefore an invariant probability measure for a semigroup. More generally, tightness of the time averages supplies the required compactness.

Cutting and stacking constructs measure-preserving transformations from towers of equal-measure levels. Cut a tower into equal-width subcolumns, add new spacer levels, and stack the resulting columns. Map each level to the one above by a measure-preserving identification; subsequent stages extend the partial map. When uncovered and top-level measures tend to zero, the limiting map is defined modulo null sets.

The standard three-cut Chacon transformation is an invertible measure-preserving transformation built by cutting and stacking with spacer vector $(0,1,0)$ at every stage. Cut the current tower into three equal subcolumns, place one spacer above the middle, and stack left to right. With initial width $2/3$ and height one on a unit interval, level counts and widths are $h_j=(3^{j+1}-1)/2$ and $w_j=2/3^{j+1}$. Tower measure tends to one, using total spacer measure $1/3$. Tower words obey $W_{j+1}=W_j{W}_{j}1W_j$.

A continuous self-map of a compact metric space is uniquely ergodic when it has exactly one invariant Borel probability measure. An irrational rotation of the circle is uniquely ergodic: invariance multiplies its nonzero-index Fourier coefficients by nontrivial phases, forcing those coefficients to vanish. The Stone-Weierstrass theorem then identifies the invariant measure as normalized Lebesgue measure.

For a uniquely ergodic continuous map $T$ of a compact metric space, every continuous $f$ satisfies $N^{-1}{\sum }_{j=0}^{N-1}f(T^{j}x)\to \int fd\mu$ uniformly in $x$. Otherwise choose increasingly long orbit empirical measures with a fixed discrepancy. Compactness provides a weak limit; the telescoping identity makes it invariant, so it must be the unique measure $\mu$, contradicting that discrepancy. Compactness and continuity are essential hypotheses.

For a bounded sequence $(u_n)$ in a Hilbert space, put $c_h={\mathrm{limsup}}_{N\to \infty }|N^{-1}{\sum }_{n=0}^{N-h-1}⟨u_{n+h},u_n⟩|$. If $H^{-1}{\sum }_{h=1}^H{c}_h\to 0$, then $N^{-1}{\sum }_{n=0}^{N-1}{u}_n\to 0$ in norm. Since the $c_h$ are bounded, it suffices that $(c_h)$ has convergence in density of a sequence to zero. The proof applies the Cauchy-Schwarz inequality to an average of shifted sums.

For complex numbers $|u_n|\le 1$, $0\le n<N$, and $1\le H\le N$, put $C_h={\sum }_{n=0}^{N-h-1}{u}_{n+h}\overline{u_n}$. ThenExtend the sequence by zero, count each summand in $H$ consecutive windows, and apply the Cauchy-Schwarz inequality to their sum. Expansion of the squared window sums gives the stated coefficients. Fixing $H$ and making each normalized correlation tend to zero bounds the limiting squared average by $1/H$; then let $H$ tend to infinity. This is the scalar finite form underlying the Van der Corput lemma.

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is equidistributed if the limiting frequency in every half-open interval equals that interval's length. Equivalently, its empirical probability measures converge to normalized Lebesgue measure when tested against every continuous function.

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is an equidistributed sequence if and only if $N^{-1}{\sum }_{n=0}^{N-1}{e}^{2\pi im{x}_n}\to 0$ for every nonzero integer $m$. Approximation of continuous functions by trigonometric polynomials proves the criterion.

If a circle-valued sequence is not an equidistributed sequence, some positive-shift difference is not equidistributed either. The contrapositive follows by applying the Van der Corput inequality for finite scalar sequences to every nonzero integer exponential sum. Thus cancellation for all fixed nonzero differences implies equidistribution of the original sequence.

If $a$ is irrational and $b,c$ are real, the fractional parts of $a{n}^2+bn+c$ form an equidistributed sequence. In the Van der Corput lemma, every nonzero-lag correlation of $e^{2\pi im(a{n}^2+bn+c)}$ is a geometric series with irrational frequency $2mah$ and tends to zero. The Weyl criterion then applies.

A measurable transformation $T:X\to X$ is nonsingular with respect to $\mu$ when $\mu (T^{-1}N)=0$ for every $\mu$-null set $N$, equivalently when the pushforward measure $T_{\ast }\mu$ is absolutely continuous with respect to $\mu$.

For a measurable dynamical map $F$, the Koopman operator acts on observables by $K_{F}g=g\circ F$. If $F$ is nonsingular, it defines an operator on equivalence classes of measurable functions.

An $L^2$ observable of an invertible probability measure-preserving system is almost periodic when its Koopman operator orbit is totally bounded in the global $L^2$ norm. A noninvertible system uses its forward orbit. This is the Hilbert-space dynamical definition, distinguished from uniform almost periodicity for continuous functions of a real variable.

For a probability measure-preserving system, if $⟨U_T^{n}f,f⟩\to |\int fd\mu {|}^2$, then $U_T^{n}f$ converges weakly in $L^2$ to $(\int fd\mu )1$. Centre $v=f-(\int f)1$; its autocorrelations tend to zero. The isometry identity gives the same decay against each $U_T^{k}v$, hence by approximation against their closed forward span. Against its orthogonal complement the correlations are exactly zero. Orthogonal decomposition by a closed subspace finishes the proof without invertibility.

For a nonsingular map $F$ on a probability space, $K_F$ extends boundedly to $L^2$ exactly when $d(F_{\ast }\mu )/d\mu \in L^{\infty }$. In that case

A measurable transformation $T:X\to X$ preserves a measure $\mu$ when $\mu (T^{-1}A)=\mu (A)$ for every measurable set $A$.

Such a measurable self-map preserves Lebesgue measure. On the unit interval examples include the identity, reflection $t\mapsto 1-t$, and $t\mapsto 2t(\mathrm{mod}1)$. Invertibility is not required.

For an integer $K\ge 2$, multiplication by $K$ on the circle group preserves normalized Lebesgue measure. Each inverse image consists of $K$ branches, each scaling lengths by $1/K$.

The integer multiplication map on the circle is an ergodic transformation for Lebesgue measure. In the Fourier basis, an invariant function has coefficients $c_j=0$ when $K$ does not divide $j$, and $c_j=c_{j/K}$ otherwise. Every nonzero index eventually reduces to the first case, so only the constant coefficient remains.

A measure-preserving system is a measure space $(X,\mathcal{B},\mu )$ together with a measure-preserving transformation $T$. In ergodic theory the measure is commonly a probability measure.

A factor map $\pi$ is a measurable map satisfying $\pi \circ T=S\circ \pi$ and ${\pi }_{\ast }\mu =\nu$, from one measure-preserving system to another. The target system is the factor; its observable functions pull back to observable functions of the source.

A measurable measure-preserving map intertwining the transformations of two probability measure-preserving systems. The equalities may be imposed on invariant full-measure domains. It represents the target's observable information inside the source. The conditional measures of a factor describe the information remaining on each fiber.

An extension for which the relatively almost periodic observables are dense in global $L^2$. Density does not assert that every $L^2$ function already meets the uniform finite-net condition on fibers. Localization yields the positive-measure almost periodic indicator in a compact extension.

For a skew-product extension with finitely many labeled fibers, the orbit of a bounded observable on each fiber lies in a fixed bounded subset of a finite-dimensional vector space. A finite net there supplies global centers constant in the base coordinate. Bounded observables are dense in $L^2$, so this is a compact extension. Global almost periodicity can still fail because the base can be weakly mixing.

Approximate $1_A$ by relative almost periodic functions with summable squared global errors. Tonelli theorem makes their conditional errors tend to zero almost everywhere. Choose a positive base set on which the conditional mass of $A$ is bounded below, then use the Egorov theorem to make those errors uniform on a smaller positive set $C$. Localization of relative almost periodicity to base sets and conditional-norm closure prove $1_B$ is relatively almost periodic and $\mu (B)>0$.

For every positive $\epsilon$, require finitely many global $L^2$ centers $g_j$ such that the displayed inequality holds for every integer $n$ on almost every fiber of the factor map. A center can depend on the fiber and time; its finite list is fixed. This is weaker than almost periodic observable in global norm. For noninvertible systems use nonnegative times.

Multiplying a relatively almost periodic observable by a base-set indicator preserves relative almost periodicity: on each fiber an orbit value is either unchanged or zero, so add zero to the finite list of centers. Conditional norm covariance under a factor map also proves closure under convergence in the uniform conditional L2 norm.

Let $X$ be a finite set, let every singleton have positive probability, and let $T:X\to X$ preserve that probability measure. For every $x\in X$,so every $x$ has a preimage. Thus $T$ is a surjection, and every surjection from a finite set to itself is a bijection.

An invertible measure-preserving system has a measurable inverse modulo null sets and preserves its measure in both time directions. Its Koopman operator on $L^2$ is unitary.

The invariant sigma-algebra of $T$ consists, modulo null sets, of measurable sets $A$ satisfying $T^{-1}A=A$.

A measurable set $A$ satisfying $T^{-1}A=A$, or equality modulo null sets in the measure-theoretic convention. These sets form a $\sigma$-algebra, and an ergodic measure-preserving transformation gives each invariant set probability zero or one.

A measure-preserving transformation is ergodic when every set in its invariant sigma-algebra has measure zero or full measure. Equivalently, every integrable invariant function is almost everywhere constant.

A probability measure-preserving transformation is strongly mixing when $\mu (T^{-n}A\cap B)\to \mu (A)\mu (B)$ for every pair of measurable sets $A,B$. Unlike weak mixing, this is an ordinary limit along all sufficiently large times.

A probability measure-preserving transformation is strong mixing exactly when $\mu (T^{-n}A\cap A)\to \mu (A{)}^2$ for every measurable $A$. The converse follows by applying decay of autocorrelation implies weak convergence of an observable to $f=1_A$ and then testing against $1_B$. Diagonal polarization alone controls a sum of the two directed cross-correlations and does not suffice as a proof.

For a measure-preserving system on a standard probability space, an invariant probability measure is an integral of ergodic invariant probability measures. Its conditional measures given the invariant sigma-algebra are the ergodic components.

For rotation by $p/q$ with coprime integers and Haar probability on the circle, components are the equal measures on the $q$-point orbits. The quotient map is $x\mapsto qx(\mathrm{mod}1)$. Integration over the quotient recovers Haar measure, and the rotation permutes each orbit transitively, making its probability measure ergodic.

A conditional probability of an invariant probability measure given its invariant sigma-algebra. In a standard Borel probability system, almost every such probability is invariant and ergodic. Their integral recovers the original measure. The countable-test proof of ergodicity of conditional components explains why invariance alone is not the whole conclusion.

Apply the Birkhoff ergodic theorem to a countable generating algebra, and disintegrate the resulting common full-measure convergence set over the invariant sigma-algebra. The orbit limits on a fiber are its conditional masses. Applying the theorem to that invariant component shows its invariant projection is constant on all generating indicators, hence on all $L^2$ functions by density. This proves ergodicity without an uncountable intersection of null sets.

If $\mu =\int \nu d\tau (\nu )$ is an ergodic decomposition, then the Kolmogorov-Sinai entropy satisfies $h_{\mu }(T)=\int h_{\nu }(T)d\tau (\nu )$.

Let $T$ be an ergodic transformation preserving a probability measure $\mu$ on a countable set. Some state $x$ has $\mu (\{x\})>0$. Along its forward orbit,An infinite sequence of distinct states with masses bounded below by $\mu (\{x\})$ is impossible, so the orbit eventually reaches a finite cycle $Y$. Since $Y\subseteq T^{-1}Y$ and invariance gives $\mu (T^{-1}Y)=\mu (Y)>0$, the difference is null. The cycle is invariant modulo null sets, and ergodicity therefore gives $\mu (Y)=1$.

An invertible measure-preserving system is ergodic exactly when the fixed space of its Koopman operator on $L^2$ consists only of almost-everywhere constant functions. One direction applies invariance to level sets; the other applies it to indicator functions of invariant sets.

On a one-sided sequence space $E^{\mathbb{N}}$ with product measure, the Bernoulli shift isIt preserves the product measure. Every invariant event lies in the tail sigma-algebra of the coordinate process, so the Kolmogorov zero-one law makes the shift an ergodic transformation.

For independent fair signs on a two-sided Bernoulli shift, the coordinate observable $f(y)=y_0$ has unit norm and pairwise orthogonal orbit values. Their pairwise squared distance is two, excluding a finite small net in global $L^2$. Its pullback along a factor map can nevertheless be a relatively almost periodic observable, since it is fiber-constant and takes only two values.

The law of ${\sum }_{j\ge 1}2{\omega }_j{3}^{-j}$, for independent fair binary digits ${\omega }_j$, is supported on the Cantor set. It is invariant and ergodic under $T_3$, as the image of a Bernoulli shift. Each permitted ternary cylinder of length $n$ has measure $2^{-n}$, giving entropy rate $\log 2$. The Host equidistribution theorem makes almost every point a normal number in base $2$, while its ternary digits omit $1$.

For irrational $\alpha$, the circle rotation $T(x)=x+\alpha (\mathrm{mod}1)$ preserves Lebesgue measure and is an ergodic transformation.

The rotation $x\mapsto x+a$ of a circle of circumference $2\pi$ is ergodic exactly when $a/\pi$ is irrational. In the Fourier basis, an invariant function can have a nonzero $k$th coefficient only when $e^{ika}=1$.

Every orbit of an irrational rotation of the circle visits a half-open interval $I$ with limiting frequency equal to its length. One proof sandwiches the orbit of an arbitrary point between inner and outer intervals along a nearby orbit for which the Birkhoff ergodic theorem holds.

A probability measure-preserving transformation $T$ is weakly mixing when, for every pair of measurable sets $A,B$, the average absolute correlation discrepancy tends to zero:Equivalently, the correlations converge to their products in convergence in density of a sequence, its Cartesian product with itself is an ergodic transformation, or its product with every ergodic transformation is ergodic. The absolute values are essential: signed Cesaro convergence of a sequence of the correlations to their products characterizes ergodic transformations and does not imply weak mixing. For example, the transformation interchanging two equally weighted points has discrepancy $(-1{)}^n/4$ for a singleton $A=B$: its signed average tends to zero while its average absolute discrepancy stays $1/4$.

For bounded observables of a probability weak mixing system, the displayed convergence holds in $L^2$. Induct on $k$: center the final observable, compute the lagged Hilbert-space correlations, apply the induction hypothesis to the $k-1$ remaining factors, and use weak mixing of $T^k$ with the Van der Corput lemma to obtain zero. Adding back the final mean completes the induction.

For real bounded observables, the Cesaro limit of the correlation comes from arithmetic-progression multiple averages under weak mixing. Apply the same result on the weakly mixing product system to $f_j\otimes f_j$ to obtain its second moment. The averaged squared error tends to zero, so the mean-square criterion for convergence in density gives the displayed density limit.

Every positive power of a weak mixing transformation is weakly mixing: its nonnegative discrepancies are a subsequence whose average is bounded by $r$ times the full average. Products are weakly mixing because correlations of simple tensors factor; the absolute product discrepancy is bounded by the sum of bounded multiples of the single-system discrepancies. Density of finite tensor sums gives the full $L^2$ assertion.

If $T$ is weakly mixing, then $T\times S$ is ergodic exactly when $S$ is ergodic. In particular, ergodicity of $T\times T$ is equivalent to weak mixing of $T$.

For mean-zero $f$, apply the mean ergodic theorem on an ergodic product system to $f\otimes \overline{f}$, paired with $g\otimes \overline{g}$. Its correlation is $|⟨U^{n}f,g⟩{|}^2$. The Cauchy-Schwarz inequality then gives averaged absolute correlation decay, the definition of weak mixing.

A weakly mixing system has no nonconstant Koopman operator eigenfunction. If $U_{T}f=\lambda f$, then $f(x)\overline{f(y)}$ is invariant under $T\times T$, contradicting the product characterization of weak mixing unless $f$ is constant.

A system is weakly mixing exactly when, for every positive-measure $A,B,C$, some $n\ge 1$ makes both $T^{-n}A\cap B$ and $T^{-n}A\cap C$ have positive measure. This is the finite simultaneous-hitting form of the density-one correlation characterization.

In a finite measure-preserving system, almost every point of every measurable set $A$ returns to $A$ infinitely often. The points of $A$ that never return form a wandering set: all its inverse images are pairwise disjoint and have equal measure, so that set must be null.

For a probability measure-preserving system, every measurable $B$ of positive measure and every integer $k\ge 2$ admit an integer $n\ge 1$ with $\mu ({\bigcap }_{j=0}^{k-1}{T}^{-jn}B)>0$. The sets are preimages; neither invertibility nor an ergodic transformation is required. The Furstenberg correspondence principle converts this simultaneous return into arithmetic progressions and yields the Szemerédi theorem. The case $k=2$ follows from the Poincare recurrence theorem.

Every circle rotation has the Furstenberg multiple recurrence theorem property for normalized Lebesgue measure. For a rational angle, use a period. For an irrational angle, choose a positive return time with angle close to zero; translation continuity in L1 on the circle then makes finitely many translates of a given positive-measure set simultaneously close to that set, and the union bound leaves a positive-measure intersection.

For a measure-preserving transformation $T$ and $f\in L^1(\mu )$, the ergodic averagesconverge almost everywhere to $\mathbb{E}[f\mid \mathcal{I}]$, the conditional expectation on the invariant sigma-algebra. The limit has the same integral as $f$.

In a probability measure-preserving system whose transformation is an ergodic transformation, a nonnegative measurable $f$ with $\int fd\mu =+\infty$ has $N^{-1}{\sum }_{j=0}^{N-1}f(T^{j}x)\to +\infty$ almost everywhere. Apply the pointwise ergodic theorem to $\min (f,K)$ on a common full-measure set for integer $K$, then bound the lower limit by each truncated integral. The monotone convergence theorem sends those integrals to infinity. The finite-integral case is the ordinary pointwise ergodic theorem.

For a measure-preserving transformation and real integrable $f$, let $S_{k}f={\sum }_{j<k}f\circ T^j$. For $E_N=\{{\max }_{1\le k\le N}{S}_{k}f>0\}$ the lemma asserts ${\int }_{E_N}fd\mu \ge 0$. Passing to the increasing union gives the corresponding infinite-supremum inequality.

Suppose $f_k\to f$ almost everywhere and ${\sup }_k|f_k|\in L^1$ in a probability measure-preserving system. Thenalmost everywhere and in $L^1$, where $\mathcal{I}$ is the invariant sigma-algebra. For the almost-everywhere assertion, bound the terms with index at least $M$ by ${\sup }_{k\ge M}|f_k-f|$, apply the Birkhoff ergodic theorem, and then let $M\to \infty$. The finitely many remaining end terms vanish by the linear growth bound for integrable observables.

In a probability measure-preserving system, $f\in L^1$ implies $f(T^{n}x)/n\to 0$ almost everywhere. For every $\epsilon >0$, the tail integral formula for moments gives ${\sum }_{n\ge 1}\mu (|f|>\epsilon n)\le \Vert f{\Vert }_1/\epsilon$. Invariance and the Borel-Cantelli lemma finish the proof. Consequently division by $n^a$ also gives zero for every $a\ge 1$.

For every $0<a<1$, choose $1<p<1/a$. On the Bernoulli shift over independent uniform coordinates in $(0,1)$, the observable $f(x)=x_0^{-1/p}$ is integrable, but $\mu (f(T^{n}x)>n^a)=n^{-ap}$. The events are independent and their probabilities have divergent sum, so the Borel-Cantelli lemma implies that $f(T^{n}x)/n^a>1$ infinitely often almost surely. No exponent below one gives a universal bound for all integrable observables.

On a finite measure space, the convergence in the Birkhoff ergodic theorem also holds in $L^1$. Prove it first for bounded truncations by the dominated convergence theorem, then use the $L^1$ contraction $\Vert A_{n}h{\Vert }_1\le \Vert h{\Vert }_1$ and Fatou lemma to remove the truncation.

For $S\subseteq \mathbb{Z}$, its upper Banach density is

The Furstenberg correspondence principle turns a subset $S\subseteq \mathbb{Z}$ of positive upper Banach density into a shift-invariant probability system and a positive-measure cylinder $A$ such that recurrence times of $A$ produce differences of elements of $S$.

A sequence $a_n$ converges in density to $a$ when, for every $\epsilon >0$,

The displayed condition implies convergence in density of a sequence to $\ell$, since the fraction of indices with $|a_n-\ell |\ge \epsilon$ is at most ${\epsilon }^{-2}$ times the averaged squared error. A signed Cesaro limit alone is insufficient: $a_n=(-1{)}^n$ has zero signed average but no density limit.

A sequence $a_n$ converges in the Cesaro sense to $a$ when $N^{-1}{\sum }_{n=1}^N{a}_n\to a$. For bounded sequences, convergence in density to $a$ is equivalent to Cesaro convergence of $|a_n-a|$ to zero.

For a finite measurable partition $\xi$, its entropy iswith $0\log 0=0$.

For finite partitions $\xi ,\eta$,Conditioning reduces entropy, so $H_{\mu }(\xi \mid \eta )\le H_{\mu }(\xi )$.

The entropy rate of $\xi$ isThe numerator is a subadditive sequence, so the limit exists and equals the infimum of the displayed ratios.

For a finite measurable partition $\xi$, put ${\mathcal{F}}_1=\sigma ({\bigvee }_{j\ge 1}{T}^{-j}\xi )$. Then $h_{\mu }(T,\xi )=H_{\mu }(\xi \mid {\mathcal{F}}_1)={\lim }_{n\to \infty }{H}_{\mu }(\xi \mid {\xi }_1^n)$. The backward entropy chain rule writes $H_{\mu }({\xi }_0^{N-1})$ as the sum of the first $N$ decreasing finite-future conditional entropies. Their averages have the same limit. Increasing conditioning fields converge by the Martingale convergence theorem.

For ${\mathcal{F}}_L=\sigma ({\bigvee }_{j\ge L}{T}^{-j}\xi )$ and a finite measurable partition $\xi$, $H_{\mu }({\xi }_0^{L-1}\mid {\mathcal{F}}_L)=L{h}_{\mu }(T,\xi )$. The backward conditional entropy chain rule expresses the left side as ${\sum }_{j=0}^{L-1}{H}_{\mu }(T^{-j}\xi \mid {\mathcal{F}}_{j+1})$. Each term equals $h_{\mu }(T,\xi )$ by measure preservation and the infinite-future formula for partition entropy rate. No invertibility is needed.

For a probability measure-preserving system and a countable measurable partition $\xi$ of finite entropy, $-n^{-1}\log \mu ({\xi }_0^{n-1}(x))$ converges almost everywhere and in $L^1$ to an invariant function whose integral is $h_{\mu }(T,\xi )$. For an ergodic transformation, the limit is the constant $h_{\mu }(T,\xi )$. In general it is $\mathbb{E}[I_{\mu }(\xi \mid {\bigvee }_{j=1}^{\infty }{T}^{-j}\xi )\mid \mathcal{I}]$. The Martingale convergence theorem, maximal inequality for conditional information functions, and triangular ergodic averaging lemma prove this directly, without invertibility.

For a stationary process associated with a finite-entropy measurable partition, $H_{\mu }({\xi }_0^{n-1})/n$ is non-increasing. The increments $c_k=H_{\mu }(\xi \mid {\bigvee }_{j=1}^k{T}^{-j}\xi )$ are non-increasing by conditioning reduces entropy, and the chain rule gives $H_{\mu }({\xi }_0^{n-1})={\sum }_{k=0}^{n-1}{c}_k$. Averages of a decreasing sequence are decreasing.

The Kolmogorov-Sinai entropy is $h_{\mu }(T)={\sup }_{\xi }{h}_{\mu }(T,\xi )$ over finite measurable partitions.

A probability measure-preserving system has completely positive entropy if $h_{\mu }(T,\xi )>0$ for every finite measurable partition with $H_{\mu }(\xi )>0$. Equivalently, its Pinsker sigma-algebra is trivial. Since finite partitions measurable in a partition tail have zero entropy rate, this property forces every finite-partition tail sigma-algebra of a measurable partition to be trivial.

For an invertible measure-preserving system, a generating measurable partition $\xi$ has $\sigma ({\bigvee }_{j\in \mathbb{Z}}{T}^{-j}\xi )=\mathcal{B}$ modulo null sets. Thus knowing the entire two-sided itinerary determines all measurable observations. A finite or countable finite-entropy generating partition computes the system entropy through the Kolmogorov-Sinai generator theorem.

A one-sided generator is a measurable partition $\xi$ with $\sigma ({\bigvee }_{j\ge 0}{T}^{-j}\xi )=\mathcal{B}$ modulo null sets. Equivalently, every measurable set can be approximated in measure by unions of atoms of finite forward-name blocks. A one-sided coordinate partition generates a one-sided Bernoulli shift; for a nontrivial base distribution, it does not generate the two-sided version, whose negative coordinates are independent of the nonnegative ones.

If an invertible probability measure-preserving system has a finite one-sided generator $\xi$, then its future sigma-algebra ${\mathcal{F}}_1=T^{-1}\mathcal{B}$ equals $\mathcal{B}$ modulo null sets. Hence $H_{\mu }(\xi \mid {\mathcal{F}}_1)=0$, and the infinite-future formula for partition entropy rate and Kolmogorov-Sinai generator theorem give $h_{\mu }(T)=0$. Invertibility matters: a fair binary one-sided Bernoulli shift has entropy $\log 2$ and a finite one-sided generator.

For an invertible probability measure-preserving system with a finite or countable finite-entropy generating measurable partition $\xi$, $h_{\mu }(T)=h_{\mu }(T,\xi )$. For a noninvertible system the same conclusion holds for a finite-entropy one-sided generator. The finite-entropy hypothesis is part of this form of the theorem.

Let $p,q\ge 2$ be relatively prime integers. If $\nu$ is invariant and ergodic for the integer multiplication map on the circle $T_p$ and $h_{\nu }(T_p)>0$, then for $\nu$-almost every $x$ the sequence $(T_q^{n}x)$ is an equidistributed sequence for Lebesgue measure. For a non-ergodic $T_p$ invariant measure, the same conclusion holds if almost every measure in its ergodic decomposition has positive entropy. Positive entropy of the whole measure alone does not suffice.

A probability measure on the circle invariant under $T_2$ and $T_3$ and ergodic for their jointly generated semigroup is Lebesgue measure if either transformation has positive Kolmogorov-Sinai entropy. Without joint ergodicity, a mixture of Lebesgue measure and an atomic invariant measure shows why positive entropy alone is insufficient.

For standard probability systems, a factor of a measure-preserving system with at most $q$ points in every fibre preserves Kolmogorov-Sinai entropy. Conditional on the complete factor point, every finite orbit name has at most $q$ possibilities, so its conditional entropy is at most $\log q$. Dividing by the orbit length gives zero relative entropy.

The Pinsker sigma-algebra consists of the measurable sets $A$ whose binary partition $\{A,X\setminus A\}$ has entropy rate zero. It is a sigma-algebra and is the largest invariant factor with zero Kolmogorov-Sinai entropy.

A set belongs to the Pinsker sigma-algebra exactly when, modulo a null set, it belongs to $\mathcal{T}(\xi )$ for some finite measurable partition $\xi$. For a zero-entropy binary partition one may take that partition itself: zero conditional entropy makes its present atom measurable from every remote future.

For a measure-preserving action of $\mathbb{Z}$ and a finite partition $\xi$,The reverse inequality to the interval formula follows from Shearer's inequality applied to translates of $F$ along long intervals; the uncovered boundary has sublinear size.

A system is K-mixing when every finite partition becomes asymptotically independent of every event in its remote future: if ${\mathcal{F}}_n(\xi )={\bigvee }_{k=n}^{\infty }{T}^{-k}\xi$, thenfor every measurable $A$ and every finite $\xi$.

The tail sigma-algebra of a finite partition $\xi$ isBy the reverse martingale convergence theorem, K-mixing is equivalent to triviality of $\mathcal{T}(\xi )$ for every finite partition $\xi$.

Let $\alpha$ be finite and measurable in $\mathcal{T}(\xi )$ for a finite measurable partition $\xi$, and put $h=h_{\mu }(T,\xi )$. Choose $r$ with $H(\alpha \mid {\xi }_0^r)<\epsilon$. For $\gamma ={\alpha }_0^{n-1}$ and $\beta ={\xi }_0^{n+r-1}$, conditional subadditivity gives $H(\gamma \mid \beta )<n\epsilon$. Tail measurability and block conditional entropy given the infinite future give $H(\beta \mid \gamma )\ge (n+r)h$. Thus $H(\gamma )\le H(\beta )-(n+r)h+n\epsilon$. Divide by $n$, then let $n\to \infty$ and $\epsilon \downarrow 0$, proving $h_{\mu }(T,\alpha )=0$. The proof also works for noninvertible transformations.