Correlation Number for Potentials with Entropy Gaps and Cusped Hitchin Representations

We introduce a correlation number for two strictly positive, locally Hölder 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.

💡 Research Summary

The paper introduces a new invariant, the “correlation number,” for pairs of strictly positive, locally Hölder continuous potentials on a countable‑state, topologically mixing Markov shift with the big‑images‑and‑pre‑images (BIP) property. The potentials are assumed to have strong entropy gaps at infinity, meaning that the series
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