Correlation Number for Potentials with Entropy Gaps and Cusped Hitchin Representations

TL;DR

This paper introduces a correlation number for certain potentials on Markov shifts and explores its connection to Hitchin representations and the Manhattan curve, revealing rigidity properties.

Contribution

It defines a new correlation number for potentials with entropy gaps and links it to cusped Hitchin representations and the Manhattan curve.

Findings

Correlation number is well-defined for potentials with entropy gaps.

Established a connection between the correlation number and the Manhattan curve.

Identified rigidity properties of the correlation number.

Abstract

We introduce a correlation number for two strictly positive, locally H\"older continuous, independent potentials with strong entropy gaps at infinity on a topologically mixing countable state Markov shift with BIP. We define in this way a correlation number for pairs of cusped Hitchin representations. Furthermore, we explore the connection between the correlation number and the Manhattan curve, along with several rigidity properties of this correlation number.