Measure-preserving transformation ($T$)

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.