Thermodynamic formalism and $k$-bonacci substitutions

TL;DR

This paper investigates the thermodynamic formalism of $k$-bonacci substitutions, defining a renormalization operator, analyzing pressure functions, and proving a freezing phase transition with a unique ergodic measure.

Contribution

It introduces a renormalization operator for $k$-bonacci substitutions and proves the existence of a freezing phase transition in the associated thermodynamic formalism.

Findings

Existence of a freezing phase transition in the pressure function.

Unique ergodic measure realizes the phase transition.

Analysis of the renormalization operator's iterates over potentials.

Abstract

We study $k$-bonacci substitutions. For each we define a renormalization operator associated to it and examine its iterates over potentials in a certain class. We also study the pressure function associated to potentials in this class and prove the existence of a freezing phase transition which is realized by the only ergodic measure on the subshift associated to the substitution.