Thermodynamic formalism and integral means spectrum of asymptotic tracts for transcendental entire functions

TL;DR

This paper develops a comprehensive thermodynamic formalism for a broad class of entire functions, introducing an integral means spectrum for logarithmic tracts to analyze fractal boundary behavior and derive dynamical properties.

Contribution

It introduces an integral means spectrum for logarithmic tracts, extending thermodynamic formalism to new classes of entire functions with fractal boundaries.

Findings

Established thermodynamic formalism for a wide class of entire functions.

Derived exponential decay of correlations and central limit theorem.

Provided a Bowen's formula for Hausdorff dimension of Julia sets.

Abstract

We provide the full theory of thermodynamic formalism for a very general collection of entire functions in class $\mathcal B$. This class overlaps with the collection of all entire functions for which thermodynamic formalism has been so far established and contains many new functions. The key point is that we introduce an integral means spectrum for logarithmic tracts which takes care of the fractal behavior of the boundary of the tract near infinity. It turns out that this spectrum behaves well as soon as the tracts have some sufficiently nice geometry which, for example, is the case for quasicircle, John or H\"older tracts. In this case we get a good control of the corresponding transfer operators, leading to full thermodynamic formalism along with its applications such as exponential decay of correlations, central limit theorem and a Bowen's formula for the Hausdorff dimension of radial Julia sets. Our approach applies in particular to every hyperbolic function from any Eremenko-Lyubich analytic family of Speiser class $\mathcal S$ provided this family contains at least one function with H\"older tracts. The latter is, for example, the case if the family contains a Poincar\'e linearizer.