🔄Ergodic Theory Unit 2 – Measure-Preserving Transformations & Ergodicity

Key Concepts & Definitions

• Measure space $(X, \mathcal{B}, \mu)$ consists of a set $X$, a $\sigma$-algebra $\mathcal{B}$ of subsets of $X$, and a measure $\mu$ defined on $\mathcal{B}$
• Transformation $T: X \to X$ maps elements of the set $X$ to itself
• $T$ is measurable if for every $B \in \mathcal{B}$, the preimage $T^{-1}(B) \in \mathcal{B}$
• $T$ is measure-preserving if for every $B \in \mathcal{B}$, $\mu(T^{-1}(B)) = \mu(B)$
  • Intuitively, measure-preserving transformations do not change the measure of sets under the transformation
• Ergodicity is a property of measure-preserving transformations where the only invariant sets under $T$ are those with measure 0 or 1
• Invariant set $A \in \mathcal{B}$ satisfies $T^{-1}(A) = A$ up to a set of measure zero
• Ergodic theorem relates the time average of a function along the orbit of a transformation to its space average

Measure-Preserving Transformations Explained

• Measure-preserving transformations (MPTs) are fundamental objects in ergodic theory that preserve the measure of sets under the transformation
• For a measure space $(X, \mathcal{B}, \mu)$ and a measurable transformation $T: X \to X$, $T$ is measure-preserving if $\mu(T^{-1}(B)) = \mu(B)$ for all $B \in \mathcal{B}$
• Intuitively, MPTs do not change the "size" or "volume" of sets when applying the transformation
• MPTs can be thought of as "volume-preserving" or "mass-preserving" transformations in the context of the measure space
• Examples of MPTs include rotations on the unit circle, shifts on the space of sequences, and certain billiard ball motions
• MPTs play a crucial role in understanding the long-term behavior of dynamical systems and the distribution of orbits
• The study of MPTs leads to important concepts such as ergodicity, mixing, and entropy in dynamical systems

Properties of Measure-Preserving Transformations

• Composition of MPTs: If $T$ and $S$ are MPTs on $(X, \mathcal{B}, \mu)$, then their composition $T \circ S$ is also an MPT
• Inverse of MPTs: If $T$ is an invertible MPT, then its inverse $T^{-1}$ is also an MPT
• Invariance of measure: For any measurable set $A \in \mathcal{B}$, $\mu(T^{-1}(A)) = \mu(A)$
• Preservation of null sets: If $\mu(A) = 0$, then $\mu(T^{-1}(A)) = 0$
• Preservation of sets of finite measure: If $\mu(A) < \infty$, then $\mu(T^{-1}(A)) = \mu(A) < \infty$
• Recurrence: For almost every $x \in X$, there exists a subsequence $\{n_k\}$ such that $T^{n_k}(x) \to x$ as $k \to \infty$
  • Poincaré recurrence theorem states that almost every point returns arbitrarily close to its initial position infinitely often under the action of an MPT

Introduction to Ergodicity

• Ergodicity is a fundamental concept in ergodic theory that characterizes the "irreducibility" or "indecomposability" of a measure-preserving transformation
• A measure-preserving transformation $T$ on $(X, \mathcal{B}, \mu)$ is ergodic if the only invariant sets under $T$ are those with measure 0 or 1
  • Invariant set $A \in \mathcal{B}$ satisfies $T^{-1}(A) = A$ up to a set of measure zero
• Intuitively, an ergodic transformation "mixes" the space so that any measurable set eventually spreads evenly throughout the entire space under repeated applications of the transformation
• Ergodicity implies that the space cannot be decomposed into two or more non-trivial invariant subsets
• Examples of ergodic transformations include irrational rotations on the unit circle and the doubling map on the unit interval
• Non-examples of ergodic transformations include periodic transformations and transformations with non-trivial invariant subsets
• Ergodicity is a stronger property than transitivity and implies that the transformation exhibits "chaotic" behavior

Ergodic Theorems and Their Applications

• Ergodic theorems are powerful results that relate the time average of a function along the orbit of a transformation to its space average
• Birkhoff's ergodic theorem: For an ergodic measure-preserving transformation $T$ on $(X, \mathcal{B}, \mu)$ and any integrable function $f \in L^1(X, \mu)$, the time average $\frac{1}{n} \sum_{k=0}^{n-1} f(T^k(x))$ converges almost everywhere to the space average $\int_X f d\mu$
  • Intuitively, the time average of a function along the orbit of almost every point converges to the space average of the function
• von Neumann's mean ergodic theorem: For a unitary operator $U$ on a Hilbert space $H$ and any $x \in H$, the time average $\frac{1}{n} \sum_{k=0}^{n-1} U^k x$ converges in the norm topology to the projection of $x$ onto the subspace of $U$-invariant elements
• Ergodic theorems have applications in statistical mechanics, where they relate the time average of observables to their ensemble average
• Ergodic theorems also have applications in number theory, particularly in the study of arithmetic progressions and the distribution of prime numbers
• In dynamical systems, ergodic theorems provide a way to understand the long-term behavior of orbits and the equidistribution of points

Examples and Counterexamples

• Example of an ergodic transformation: Irrational rotation on the unit circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ defined by $T_{\alpha}(x) = x + \alpha \mod 1$, where $\alpha$ is an irrational number
  • The only invariant sets under $T_{\alpha}$ are those with measure 0 or 1
• Example of a non-ergodic transformation: Rotation by a rational angle on the unit circle
  • The orbit of any point is finite and forms a non-trivial invariant set
• Example of a non-ergodic transformation: The identity transformation $T(x) = x$ on any measure space
  • Every measurable set is invariant under the identity transformation
• Counterexample to the converse of the ergodic theorem: There exists a non-ergodic transformation $T$ and an integrable function $f$ such that the time average of $f$ converges almost everywhere to the space average of $f$
• Example of a transformation that is measure-preserving but not ergodic: The transformation $T(x) = -x$ on the interval $[-1, 1]$ with the Lebesgue measure
  • The sets $[-1, 0]$ and $[0, 1]$ are non-trivial invariant sets under $T$

Connections to Other Math Fields

• Ergodic theory has strong connections to other areas of mathematics, including dynamical systems, functional analysis, and probability theory
• In dynamical systems, ergodic theory provides a framework for studying the long-term behavior of measure-preserving transformations and the distribution of orbits
  • Concepts such as mixing, weak mixing, and entropy are closely related to ergodicity
• In functional analysis, ergodic theorems can be formulated in terms of unitary operators on Hilbert spaces
  • The mean ergodic theorem of von Neumann is a fundamental result in this context
• In probability theory, ergodic theory is used to study stationary processes and the convergence of random variables
  • The ergodic theorem for stationary processes relates the time average of a process to its ensemble average
• Ergodic theory also has applications in physics, particularly in statistical mechanics and the study of thermodynamic systems
  • The ergodic hypothesis in statistical mechanics states that the time average of an observable in a system equals its ensemble average
• Ergodic theory has connections to number theory, particularly in the study of arithmetic progressions and the distribution of prime numbers
  • The ergodic theorem has been used to prove results about the density of arithmetic progressions in the integers

Problem-Solving Strategies

• When proving a transformation is measure-preserving, show that the preimage of any measurable set has the same measure as the original set
  • Use the definition of measure-preserving transformations and the properties of the measure
• To prove a transformation is ergodic, show that the only invariant sets are those with measure 0 or 1
  • Use the definition of ergodicity and the properties of invariant sets
• When applying the ergodic theorem, identify the measure-preserving transformation, the function of interest, and the space on which they are defined
  • Check that the transformation is ergodic and the function is integrable
• To find invariant sets or measures, look for subsets or functions that are preserved under the action of the transformation
  • Use the properties of invariant sets and the definition of invariant measures
• When studying the long-term behavior of a transformation, consider the convergence of time averages and the distribution of orbits
  • Apply the ergodic theorem or other convergence results, such as the Birkhoff ergodic theorem or the von Neumann mean ergodic theorem
• To prove a transformation is mixing or weakly mixing, use the definitions and properties of these concepts in relation to ergodicity
  • Mixing implies ergodicity, but the converse is not true
• When encountering a new transformation or dynamical system, try to identify its measure-preserving and ergodic properties
  • Look for invariant sets, measures, and the behavior of orbits under the transformation