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The Hopf Decomposition is a fundamental theorem in ergodic theory, with important implications in various areas such as algebraic topology and statistical mechanics. Named after Eberhard Hopf, this theorem decomposes an invariant measure preserving transformation into a conservative, ergodic part and a completely dissipative part.
Definitions:
Given a measure space $(X, \mu)$ with an invertible, non-singular transformation $T : X \to X$, i.e. a transformation which with its inverse is measurable and carries null sets onto null sets.
- Wandering Set: A measurable subset $W$ of $X$ is wandering if its characteristic function $q = X_W$ in $A = L^\infty(X)$ satisfies $q\tau^n(q) = 0$ for all $n$; thus, up to null sets, the translates in cotangent bundle $T^n(W)$ are pairwise disjoint.
- Dissipative Action: An action is dissipative if $X = \coprod T^n(W)$ for some wandering set $W$.
- Conservative Action: An action is conservative if $X$ has no wandering subsets of positive measure.
- Incompressible Action: An action is said to be incompressible if whenever a measurable subset $Z$ satisfies $T(Z) \subseteq Z$ then $T(Z)$ has measure zero. Equivalently, if $q = \chi_Z$ and $\tau(q) \leq q$, then $\tau(q) = q$.
Formal Definition and Statement
Let $(X, \mathcal{F}, \mu, T)$ be a measure-preserving system, where $X$ is the state space, $\mathcal{F}$ is the σ-algebra of subsets of $X$, $\mu$ is a probability measure on $\mathcal{F}$, and $T$ is a measurable transformation on $X$ that preserves $\mu$.
The theorem states that if $(X, \mathcal{F}, \mu, T)$ is a measure-preserving system, then there exists a partition of $X$ into two sets, $X = X_e \sqcup X_d$, such that:
- $X_e$ is $T$-invariant, i.e., $T(X_e) = X_e$, and the restriction of $T$ to $X_e$ is ergodic.
- For any set $E \in \mathcal{F}$ with $E \subseteq X_d$, we have $\mu(T^n(E)) \to 0$ as $n \to \infty$.
This partition breaks down $X$ into an ergodic part, $X_e$, and a completely dissipative part, $X_d$.
Proof
The proof involves establishing the existence of such a partition. One common way is to define a recurrent set and show that its complement is dissipative. A point $x \in X$ is recurrent if, for any neighborhood $U$ of $x$, the set of return times to $U$ is infinite.
Define $X_e$ as the set of recurrent points, and $X_d$ as its complement. It can then be shown that these sets satisfy the properties of the Hopf Decomposition.
Applications
The Hopf Decomposition is used to better understand and characterize dynamical systems. It is a key result in the study of ergodic systems, and has applications in fields like statistical mechanics, number theory, and stochastic processes.
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