🔄Ergodic Theory Unit 9 – Ergodic Theorems for Amenable Groups

Key Concepts and Definitions

• Ergodic theory studies the long-term average behavior of dynamical systems
• Amenable groups are a class of groups that admit a finitely additive, translation-invariant probability measure
• Følner condition characterizes amenable groups using a sequence of finite sets with small boundary compared to their size
• Mean ergodic theorem states that the time average of a function converges to its space average for ergodic transformations
  • Applies to actions of amenable groups on measure spaces
• Pointwise ergodic theorem asserts the almost everywhere convergence of ergodic averages
• Invariant measures are probability measures preserved by the action of a group or transformation
• Ergodicity is a property of a measure-preserving dynamical system where every invariant set has measure 0 or 1

Historical Context and Development

• Early ergodic theorems were proved for actions of the integers (von Neumann, Birkhoff) in the 1930s
• The concept of amenability was introduced by von Neumann in 1929 in the context of the Banach-Tarski paradox
• Day (1950) and Følner (1955) provided characterizations of amenable groups using finite sets and invariant means
• Ornstein and Weiss (1980) proved the Shannon-McMillan-Breiman theorem for actions of amenable groups
  • Generalizes the classical theorem for $\mathbb{Z}$ actions
• Lindenstrauss (2001) proved pointwise ergodic theorems for tempered Følner sequences in amenable groups
• Recent developments include ergodic theorems for multiple commuting transformations and nilpotent group actions

Amenable Groups: An Overview

• Amenable groups include all finite groups, abelian groups, and solvable groups
• Non-amenable groups include free groups on two or more generators and groups containing free subgroups (Tits alternative)
• Følner condition: For every $\varepsilon > 0$ and finite set $F \subset G$, there exists a finite set $A \subset G$ such that $|FA \setminus A| < \varepsilon |A|$
  • Measures the size of the boundary of $A$ relative to its size
• Amenability is preserved under taking subgroups, quotients, extensions, and directed unions
• The class of amenable groups is closed under direct products and wreath products
• Equivalent characterizations of amenability include the existence of an invariant mean and the Reiter condition
• Amenability can be defined for semigroups and measured groupoids

Ergodic Theorems for Amenable Groups

• Mean ergodic theorem for amenable groups: Let $G$ be an amenable group acting on a Hilbert space $H$ by unitary operators. Then for any $x \in H$, the ergodic averages converge in norm to the projection of $x$ onto the space of invariant vectors
  • Generalizes von Neumann's mean ergodic theorem for $\mathbb{Z}$ actions
• Pointwise ergodic theorem for amenable groups: Let $G$ be an amenable group acting on a probability space $(X, \mu)$ by measure-preserving transformations. Then for any $f \in L^1(X, \mu)$, the ergodic averages converge almost everywhere to the conditional expectation of $f$ with respect to the $\sigma$-algebra of invariant sets
  • Extends Birkhoff's pointwise ergodic theorem for $\mathbb{Z}$ actions
• Lindenstrauss' pointwise ergodic theorem for tempered Følner sequences: Strengthens the pointwise convergence result under additional assumptions on the Følner sequence
• Ergodic theorems for multiple commuting transformations and nilpotent group actions
  • Require more sophisticated techniques and impose constraints on the group and the action

Proof Techniques and Strategies

• Transference principle: Reduces the proof of ergodic theorems for amenable groups to the case of $\mathbb{Z}$ actions
  • Allows the use of classical ergodic theorems and Fourier analysis techniques
• Følner sequences and tilings: Construct Følner sets with desirable properties (e.g., tempered, invariant under certain subgroups)
  • Used to control the error terms in the convergence of ergodic averages
• Maximal inequalities and covering lemmas: Provide bounds on the maximal function of ergodic averages
  • Help establish pointwise convergence and control exceptional sets
• Spectral theory and representation theory: Analyze the unitary representation of the group on the space of square-integrable functions
  • Used in the proof of the mean ergodic theorem and its generalizations
• Martingale convergence theorems: Employed in the proof of pointwise ergodic theorems
  • Relate the convergence of ergodic averages to the convergence of martingales

Applications in Ergodic Theory

• Ergodic Ramsey theory: Studies the existence of large structured subsets in sets of positive measure under the action of an amenable group
  • Szemerédi's theorem for amenable groups, Furstenberg-Katznelson multidimensional Szemerédi theorem
• Entropy theory: Generalizes the Shannon-McMillan-Breiman theorem and the Ornstein isomorphism theorem to actions of amenable groups
  • Classification of Bernoulli shifts over amenable groups
• Orbit equivalence and measured group theory: Investigates the relationship between the dynamical properties of group actions and the algebraic properties of the group
  • Ornstein-Weiss theorem on the orbit equivalence of free ergodic actions of amenable groups
• Rigidity phenomena: Studies the rigidity of measure-preserving actions of amenable groups
  • Zimmer's cocycle superrigidity theorem, Furstenberg's $\times 2 \times 3$ problem
• Banach space theory: Applies ergodic theorems to the study of the geometry of Banach spaces
  • Characterization of reflexive spaces using the mean ergodic theorem

Connections to Other Mathematical Fields

• Harmonic analysis: Ergodic theorems are closely related to the convergence of Fourier series and the Fourier transform on groups
  • Wiener's ergodic theorem, Bochner-Herglotz theorem
• Operator algebras: Amenability of groups is connected to the existence of invariant states on group von Neumann algebras and C*-algebras
  • Connes-Feldman-Weiss theorem on the equivalence of amenability and hyperfiniteness
• Geometric group theory: Amenability is related to the growth and isoperimetric properties of Cayley graphs
  • Følner condition and the growth of balls in Cayley graphs
• Probability theory: Ergodic theorems have probabilistic interpretations and analogues
  • Kingman's subadditive ergodic theorem, ergodic theorems for random walks on groups
• Combinatorics: Ergodic-theoretic methods are used to prove combinatorial results
  • Furstenberg's proof of Szemerédi's theorem on arithmetic progressions

Challenges and Open Problems

• Pointwise ergodic theorems for non-tempered Følner sequences: Extend Lindenstrauss' result to more general Følner sequences
• Ergodic theorems for actions of non-amenable groups: Develop analogues of ergodic theorems for actions of free groups, hyperbolic groups, and other non-amenable groups
  • Requires new ideas and techniques beyond the amenable case
• Ergodic Ramsey theory for non-amenable groups: Investigate the existence of large structured subsets in sets of positive measure under the action of non-amenable groups
  • Extends the results of ergodic Ramsey theory beyond the amenable setting
• Quantitative ergodic theorems: Obtain quantitative bounds on the rate of convergence in ergodic theorems
  • Involves the use of number-theoretic and Fourier-analytic methods
• Ergodic theorems for actions on Banach spaces: Generalize ergodic theorems to actions on Banach spaces beyond Hilbert spaces
  • Requires the development of new tools in functional analysis and operator theory
• Applications to number theory and combinatorics: Use ergodic-theoretic methods to solve open problems in number theory and combinatorics
  • Green-Tao theorem on arithmetic progressions in the primes, Erdős discrepancy problem