# The failure of Ruelle's property for entire functions

## TL;DR

This paper demonstrates that the hyperbolic dimension of certain entire functions can fail to vary analytically and explores fundamental questions in thermodynamic formalism, including the existence of hyperbolic entire functions without conformal measures.

## Contribution

It provides a counterexample to Ruelle's property for entire functions and addresses open questions in thermodynamic formalism for complex dynamics.

## Key findings

- Hyperbolic dimension does not vary analytically for a family of entire functions.
- Existence of hyperbolic entire functions without conformal measures supported on the radial Julia set.
- Counterexamples to Ruelle's property in the context of entire functions.

## Abstract

We exhibit an analytic family of hyperbolic, even disjoint type, entire functions for which the hyperbolic dimension does not vary analytically. Additionally we answer several questions in thermodynamic formalism of entire functions such as the existence of a hyperbolic entire function without conformal measure that is supported on the radial Julia set.