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

Mayer, Volker; Urbański, Mariusz

doi:10.1090/tran/8043

Thermodynamic formalism and integral means spectrum of logarithmic tracts for transcendental entire functions
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by Volker Mayer and Mariusz Urbański PDF

Trans. Amer. Math. Soc. 373 (2020), 7669-7711 Request permission

Abstract:

We provide an entirely new approach to the theory of thermodynamic formalism for entire functions of bounded type. 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 quasidisks, John, or Hölder tracts. In these cases 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.

This approach covers all entire functions for which thermodynamic formalism has been so far established and goes far beyond. It applies, in particular, to every hyperbolic function from any Erëmenko–Lyubich analytic family of Speiser class $\mathcal {S}$ provided this family contains at least one function with Hölder tracts. The latter is, for example, the case if the family contains a Poincaré linearizer.

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