5.1 Mixing and weak mixing properties

Mixing and weak mixing properties are crucial concepts in ergodic theory. They describe how measure-preserving systems behave over time, with mixing indicating stronger independence between sets than weak mixing. These properties help us understand the long-term behavior of dynamical systems.

Both mixing and weak mixing are stronger than ergodicity, forming a hierarchy of properties. They're characterized by different rates of correlation decay and spectral properties. Understanding these distinctions is key to analyzing various dynamical systems and their long-term behavior.

Mixing and Weak Mixing Properties

Definitions and Concepts

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• Mixing describes asymptotic independence of sets under the action of measure-preserving dynamical systems
• For measure-preserving transformation T on probability space (X, μ), mixing defined as $\lim_{n\to\infty} \mu(T^{-n}(A) \cap B) = \mu(A)\mu(B)$ for all measurable sets A and B
• Weak mixing uses Cesàro average of correlation function
• Weak mixing defined as $\lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N |\mu(T^{-n}(A) \cap B) - \mu(A)\mu(B)| = 0$ for all measurable sets A and B
• Mixing implies asymptotic decorrelation between measurable sets under transformation action
• L^2 functions often used instead of measurable sets for proofs and applications
• Concepts extend to measure-preserving flows (continuous-time dynamical systems) with modified definitions

Advanced Concepts and Extensions

• Koopman operator for mixing transformation has only constant functions as eigenfunctions for eigenvalue 1
• Weak mixing systems have continuous spectral measure for Koopman operator, except possible atom at 1 for constant functions
• Isomorphisms of measure-preserving systems preserve mixing and weak mixing properties
• Product of two measure-preserving transformations mixes if and only if both transformations mix
• Weak mixing equates to ergodicity of product transformation T × T on product space X × X
• Multiple mixing involves more than two sets, applied in ergodic theory and dynamical systems

Properties and Implications of Mixing

Hierarchical Relationships

• Mixing implies weak mixing, which implies ergodicity
• Implications are strict with systems existing that are weak mixing but not mixing, and ergodic but not weak mixing
• Ergodicity characterized by non-existence of non-trivial invariant sets or functions
• Mixing ensures asymptotic independence of sets, while ergodicity matches long-term average behavior to space average
• Weak mixing acts as time-averaged version of mixing, positioned between ergodicity and mixing in strength

Spectral and Correlation Properties

• Mixing systems exhibit purely continuous spectrum (except at 1)
• Weak mixing systems may have continuous spectrum with atom at 1
• Ergodic systems can possess any type of spectrum
• Correlation decay fastest in mixing systems, followed by weak mixing systems, then ergodic systems
• Strength of properties reflected in product behavior
  • Mixing preserved under finite products
  • Weak mixing preserved under countable products
  • Ergodicity may be lost even for products of two systems

Mixing vs Weak Mixing vs Ergodicity

Distinguishing Characteristics

• Ergodicity represents weakest of three properties
• Mixing systems demonstrate asymptotic independence of sets
• Weak mixing viewed as time-averaged version of mixing
• Spectral properties provide clear distinction between three concepts
• Correlation decay rates differ among mixing, weak mixing, and ergodic systems
• Product behavior varies based on strength of mixing property

Examples of Distinctions

• Irrational rotations on circle demonstrate ergodicity without weak mixing
• Certain rank-one transformations exhibit weak mixing without mixing
• Bernoulli shifts (independent symbol selection based on fixed probability distribution) exemplify mixing systems
• Gaussian systems constructed from stationary Gaussian processes create weak mixing systems without mixing property

Examples of Mixing and Weak Mixing Systems

Discrete-Time Systems

• Baker's transformation on unit square stretches horizontally and contracts vertically, exemplifying 2D mixing system
• Anosov diffeomorphisms (cat map on torus) provide mixing systems from smooth dynamics
• Rank-one transformations using cutting and stacking methods generate weak mixing transformations
  • Some rank-one transformations exhibit mixing
  • Others demonstrate weak mixing without full mixing property
• Symbolic dynamics and subshifts of finite type construct various mixing and weak mixing systems with specific properties

Continuous-Time Systems

• Flows built under function over ergodic base transformation create continuous-time mixing and weak mixing systems
• Gaussian systems based on stationary Gaussian processes yield weak mixing examples without mixing
• Anosov flows on compact manifolds provide important class of mixing continuous-time systems
• Horocycle flows on surfaces of constant negative curvature demonstrate mixing property in geometric setting