Tree pressure for hyperbolic and non-exceptional upper semi-continuous potentials

TL;DR

This paper studies the concept of tree pressure in multi-modal interval maps with specific hyperbolic and upper semi-continuous potentials, extending previous results to establish the existence of conformal measures for geometric potentials.

Contribution

It generalizes a key corollary related to tree pressure for a broader class of functions, aiding in the proof of conformal measure existence.

Findings

Generalized Corollary 2.2 for hyperbolic, non-exceptional potentials

Established existence of conformal measures for geometric potentials in negative spectrum

Extended understanding of thermodynamic formalism in dynamical systems

Abstract

In this note, we investigate the tree pressure for multi-modal interval maps with a certain class of hyperbolic and non-exceptional upper semi-continuous functions. In particular, we obtain a generalized version of Corollary 2.2 in the paper \cite{LRL14} by Li and Rivera-Letelier. This property will be used to prove the existence of a conformal measure for the geometric potential in the negative spectrum.