Mixing for free semigroup actions of Blaschke products on the circle

How quickly does a dynamical system forget its past? For analytic observables, mixing can happen at a double exponential rate, which is faster than any exponential decay. In this regime the spectrum of the associated precomposition operator is trivial, so spectral methods say nothing, and the question becomes how much of the map is remembered by the rate itself.

We will focus on finite Blaschke products of the circle with a fixed point in the unit disk, and on the free semigroup actions they generate. Blaschke products are a natural setting for this question: they are explicit and analytic, yet flexible enough to exhibit very different types of mixing behavior. For these, we will classify the possible rates: no mixing, exponential mixing, or double exponential mixing with a sharp exponent \gamma = \log p, where p \ge 2 is the minimal local degree of the generators at the fixed point. The exponent turns out to be a surprisingly non-rigid invariant. It is not stable under C^1-perturbations and does not imply C^1-conjugacy to an affine model. Rigidity appears only at the extreme: a map of degree q whose exponent attains \log q is Möbius conjugate to an affine model. Finally, we will discuss the joint spectral radius of the precomposition operators, which in the double exponential regime generate a quasi-nilpotent, non-unital algebra.