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A dynamical system is said to have the mixing property if, for any two measurable sets $A$ and $B$, the measure of their intersection after $n$ iterations of the transformation approaches the product of their individual measures as $n$ tends to infinity:

An invariant measure $\mu$ for a transformation $T$ satisfies $\mu(T^{-1}(A)) = \mu(A)$ for all measurable sets $A$. This is a fundamental concept in ergodic theory, as it ensures that the measure is not changed by the dynamics of the system.

Unique ergodicity occurs when a dynamical system has only one ergodic measure. Mathematically, a transformation $T$ is uniquely ergodic if every continuous function $f$ on the space $X$ has the same space average with respect to all $T$-invariant measures.

The ergodic decomposition theorem states that any invariant measure can be represented as an integral of ergodic measures. This means that the space of invariant measures is the convex hull of the ergodic measures and we can decompose any invariant measure into ergodic ones.

Ornstein's Isomorphism Theorem states that two Bernoulli shifts with the same entropy are isomorphic, meaning there exists a measure-preserving transformation between the two that is invertible and intertwines the shifts. This shows that entropy is a complete invariant for Bernoulli shifts.

The Poincaré Recurrence Theorem states that for a measure-preserving transformation $T$ on a finite measure space $(X, \mathcal{B}, \mu)$, almost every point in $X$ returns arbitrarily close to its initial position infinitely often. More formally, for any $A \in \mathcal{B}$ with $\mu(A) > 0$, there exists $n > 0$ such that $\mu(A \cap T^{-n}(A)) > 0$.

The Krylov-Bogolyubov Theorem provides the existence of invariant measures for continuous transformations on compact metric spaces. It states that such a system has at least one invariant probability measure, constructed as a weak limit of averages of Dirac measures along orbits of points in the space.

Weak mixing extends the concept of ergodicity and is a weaker condition than strong mixing. A transformation is weak mixing if it satisfies

The Birkhoff Ergodic Theorem states that, for an ergodic measure-preserving transformation $T$ on a probability space $(X, \mathcal{B}, \mu)$, the time average of a function $f \in L^1(\mu)$ converges to the space average almost everywhere. That is,

The Hopf Argument is used to prove the ergodicity of a system. The idea is to consider two sets, $A$ and $B$, and show that if each is invariant under the transformation, then $A$ and $B$ must have intersecting interiors unless one of them has full measure, proving ergodicity given certain conditions.

A dynamical system is ergodic if time averages equal space averages for almost every point. Mathematically, if $T: X \to X$ is a measure-preserving transformation on a probability space $(X, \mathcal{B}, \mu)$, then $T$ is ergodic if for any $A \in \mathcal{B}$, $T^{-1}(A) = A$ implies $\mu(A) = 0$ or $\mu(A) = 1$.