🔄Ergodic Theory Unit 13 – Ergodic Theory and Harmonic Analysis

Key Concepts and Definitions

• Ergodic theory studies the long-term behavior of dynamical systems with an invariant measure
• Measure-preserving transformations are functions that preserve the measure of sets in a probability space
• Ergodicity implies that the system cannot be decomposed into smaller invariant subsets and that time averages equal space averages
• Mixing is a stronger property than ergodicity and requires correlations between functions to decay over time
• Spectral theory studies the eigenvalues and eigenfunctions of the Koopman operator associated with a measure-preserving transformation
  • The Koopman operator is a linear operator that describes the evolution of observables under the dynamics
• Harmonic analysis deals with the representation of functions as superpositions of basic waves or harmonics
• Fourier analysis is a key tool in harmonic analysis and decomposes functions into their frequency components

Measure-Preserving Transformations

• A measure-preserving transformation $T: X \to X$ on a probability space $(X, \mathcal{B}, \mu)$ satisfies $\mu(T^{-1}(A)) = \mu(A)$ for all measurable sets $A$
• Measure-preserving transformations can be thought of as volume-preserving maps that do not change the total measure of sets
• Examples of measure-preserving transformations include rotations on the circle, shifts on symbolic spaces, and certain billiard maps
• The set of measure-preserving transformations forms a group under composition
• Invertible measure-preserving transformations have an inverse that is also measure-preserving
• The Poincaré recurrence theorem states that almost every point in a measure-preserving system returns to any neighborhood infinitely often
• Measure-preserving transformations can be constructed using the Rokhlin lemma, which allows the approximation of any aperiodic transformation by periodic transformations

Ergodicity and Mixing

• Ergodicity is a property of measure-preserving transformations that ensures the system cannot be decomposed into smaller invariant subsets
• A transformation $T$ is ergodic if the only invariant sets $A$ (satisfying $T^{-1}(A) = A$) have measure 0 or 1
• Ergodicity implies that time averages of functions converge to their space averages almost everywhere
  • The Birkhoff ergodic theorem formalizes this relationship between time and space averages
• Mixing is a stronger property than ergodicity and requires correlations between functions to decay over time
• A transformation $T$ is mixing if $\lim_{n \to \infty} \mu(T^{-n}(A) \cap B) = \mu(A)\mu(B)$ for all measurable sets $A$ and $B$
• Mixing implies ergodicity, but the converse is not true (irrational rotations on the circle are ergodic but not mixing)
• Mixing systems exhibit sensitive dependence on initial conditions and are often chaotic

Spectral Theory in Ergodic Systems

• Spectral theory studies the eigenvalues and eigenfunctions of the Koopman operator associated with a measure-preserving transformation
• The Koopman operator $U_T: L^2(X) \to L^2(X)$ is defined by $U_T f = f \circ T$ and describes the evolution of observables under the dynamics
• Eigenvalues of the Koopman operator are complex numbers $\lambda$ such that $U_T f = \lambda f$ for some non-zero function $f$
  • Eigenfunctions corresponding to eigenvalues are invariant under the dynamics up to a multiplicative factor
• The spectrum of the Koopman operator can be discrete (countable eigenvalues), continuous (no eigenvalues), or a combination of both
• Ergodicity is equivalent to the Koopman operator having 1 as its only eigenvalue
• Mixing is related to the absence of non-trivial eigenvalues on the unit circle
• The spectral theorem for unitary operators allows the decomposition of the Koopman operator into its discrete and continuous parts

Harmonic Analysis Fundamentals

• Harmonic analysis studies the representation of functions as superpositions of basic waves or harmonics
• The Fourier series represents periodic functions as sums of trigonometric functions (sines and cosines) with different frequencies
  • Fourier coefficients determine the amplitude of each harmonic component
• The Fourier transform extends the Fourier series to non-periodic functions and represents them as integrals of complex exponentials
• Convolution is a fundamental operation in harmonic analysis that combines two functions by integrating their product with one function shifted
  • Convolution in the time domain corresponds to multiplication in the frequency domain (convolution theorem)
• The Poisson summation formula relates the Fourier series of a periodic function to the Fourier transform of its extension
• Harmonic analysis has applications in signal processing, quantum mechanics, and partial differential equations

Fourier Analysis in Ergodic Theory

• Fourier analysis is a powerful tool in ergodic theory for studying the spectral properties of measure-preserving transformations
• The Koopman operator of an ergodic transformation has a continuous spectrum, which can be analyzed using Fourier techniques
• The spectral measure of an ergodic transformation is a measure on the unit circle that encodes the spectral properties of the Koopman operator
  • The spectral measure is related to the Fourier transform of the autocorrelation function of observables
• Mixing transformations have absolutely continuous spectral measures, while non-mixing transformations can have singular spectral measures
• The Wiener-Khinchin theorem relates the power spectrum of a stationary process to the Fourier transform of its autocorrelation function
• Fourier analysis can be used to prove limit theorems for ergodic averages, such as the central limit theorem and the law of the iterated logarithm

Applications to Number Theory

• Ergodic theory has found surprising applications in number theory, particularly in the study of arithmetic progressions and diophantine approximation
• The Furstenberg correspondence principle relates arithmetic progressions in subsets of integers to measure-preserving systems and recurrence properties
  • Furstenberg used this principle to give an ergodic-theoretic proof of Szemerédi's theorem on arithmetic progressions
• The Furstenberg-Sárközy theorem uses ergodic methods to show that polynomial sequences contain arbitrarily long arithmetic progressions
• Ergodic theory has been used to study the distribution of prime numbers and the Riemann zeta function
  • The Chowla conjecture on the correlations of the Möbius function can be formulated in terms of ergodic averages
• The Duffin-Schaeffer conjecture in diophantine approximation has been approached using ergodic-theoretic techniques
• The Lindenstrauss pointwise ergodic theorem has applications to the Littlewood conjecture on simultaneous diophantine approximation

Advanced Topics and Open Problems

• Ergodic theory has deep connections with other areas of mathematics, such as functional analysis, probability theory, and dynamical systems
• The Furstenberg-Zimmer structure theorem classifies measure-preserving systems in terms of their factors and provides a framework for studying their properties
• The Ornstein isomorphism theorem shows that certain classes of mixing transformations (Bernoulli shifts) are isomorphic and can be classified by their entropy
• The Rokhlin lemma allows the approximation of aperiodic transformations by periodic ones and is a key tool in the construction of measure-preserving transformations
• The Banach-Ruziewicz problem asks whether the Lebesgue measure is the only finitely additive, isometry-invariant measure on the sphere (solved positively in dimensions 2 and 3)
• The Rudolph-Johnson theorem characterizes the possible values of the entropy of measure-preserving transformations
• The Furstenberg-Katznelson multidimensional Szemerédi theorem extends Szemerédi's theorem to higher dimensions and has ergodic-theoretic proofs
• The Conze-Lesigne theorem on the ergodicity of certain algebraic dynamical systems has applications to the study of nilmanifolds and nilsystems