[Paper Review] Exponential mixing implies Bernoulli

Dmitry Dolgopyat, Adam Kanigowski+1 more|arXiv (Cornell University)|Jun 6, 2021

Mathematical Dynamics and Fractals33 references4 citations

TL;DR

This paper proves that for $C^{1+eta}$ diffeomorphisms on compact manifolds preserving a smooth measure, exponential mixing implies the Bernoulli property. The argument relies on exponential mixing to control correlations and uses Pesin theory and covering lemmas to establish positive Lyapunov exponents and uniform hyperbolicity, ultimately showing the system is isomorphic to a Bernoulli shift.

ABSTRACT

Let $f$ be a $C^{1+α}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $μ$. We show that if $f:(M,μ) o (M,μ)$ is exponentially mixing then it is Bernoulli.

Motivation & Objective

• To establish that exponential mixing of smooth observables implies the strongest ergodic property: Bernoullicity.
• To resolve the question of whether exponential mixing, a strong statistical property, implies stronger ergodic structure in smooth dynamical systems.
• To show that exponential mixing forces positive Lyapunov exponents and hence non-uniform hyperbolicity, a key step toward Bernoulli.
• To provide a constructive path from quantitative mixing rates to measure-theoretic isomorphism with Bernoulli shifts.
• To extend the understanding of the hierarchy of mixing and ergodic properties in smooth systems, particularly in the absence of the Rokhlin problem.

Proposed method

• Use exponential mixing estimates with $C^r$ norms to control correlations between observables under iterates of the map.
• Apply a covering argument via the Besicovitch covering theorem to find points with uniform density in the support of the measure.
• Construct smooth bump functions with controlled $C^r$ norms to probe the dynamics at small scales.
• Use the exponential mixing bound on these bump functions to show that iterates of small balls must spread and intersect distant sets.
• Establish a lower bound on the diameter of image sets under iterates, implying uniform hyperbolicity and positive Lyapunov exponents.
• Leverage Pesin theory and non-uniform hyperbolicity to conclude that the system is Bernoulli, using the fact that exponential mixing implies sufficient mixing for the isomorphism to hold.

Experimental results

Research questions

• RQ1 Does exponential mixing of smooth observables imply the Bernoulli property for $C^{1+eta}$ diffeomorphisms preserving a smooth measure?
• RQ2 Can exponential mixing be used to deduce positive metric entropy and positive topological entropy?
• RQ3 Does exponential mixing imply mixing of all orders, and if so, with what rate?
• RQ4 Is the system necessarily non-uniformly hyperbolic (i.e., has positive Lyapunov exponents) under exponential mixing?
• RQ5 Can quantitative bounds on multiple mixing be derived from exponential mixing, or is the current method non-constructive?

Key findings

• Exponential mixing of smooth observables implies the system is Bernoulli, meaning it is measure-theoretically isomorphic to a Bernoulli shift.
• The system has at least one positive Lyapunov exponent with respect to the invariant measure, provided the measure is non-atomic.
• Exponential mixing implies mixing of all orders and positive metric entropy, even though no quantitative bounds on multiple mixing are obtained.
• The proof relies on constructing test functions supported on small balls and using exponential decay of correlations to show that iterates of such balls must spread and intersect distant sets.
• The method uses a covering lemma with uniform density estimates to ensure that the measure of relevant sets is bounded away from zero, enabling the construction of hyperbolic behavior.
• The result holds under the assumption that the map is $C^{1+eta}$ and preserves a smooth measure, with the key step being the control of $C^r$ norms in the mixing estimate.