Abstract:

This work studies mixing in dynamical systems, focusing on how fast correlations between observables decay. While most previously studied systems have polynomial or exponential rates, we investigate systems with double exponential decay.

We show that all ergodic surjective linear endomorphisms of tori mix at a double exponential rate for analytic observables. In dimension one, we provide a complete classification for finite Blaschke products on the circle: the rate of mixing (no mixing, exponential, or double exponential) is determined by the value of the derivative of the Blaschke product at its fixed point. We extend the result to free semigroup actions generated by Blaschke products.

We also show that double exponential mixing is not rigid: it is not stable under perturbations and does not imply conjugacy to linear models. In higher dimensions, we construct families of examples of nonlinear maps with and without double exponential rate of mixing, and prove that certain partially hyperbolic systems never have this property.

Our approach uses the Koopman precomposition operator acting on spaces of hyperfunctions (the dual space of analytic functions). In this setting, the operator is non-self-adjoint, compact, and quasi-nilpotent, with its spectrum reduced to zero, which can be considered an indicator of double exponential decay.

Finally, we apply mixing rates to the cohomological equation. We prove the analyticity of solutions under suitable conditions and establish existence and uniqueness results in anisotropic spaces.