Tropical Markov dynamics and Cayley cubic

TL;DR

This paper explores the tropical adaptation of Markov dynamics on the Cayley cubic, revealing its semi-conjugation to a torus action and analyzing its ergodic properties, Lyapunov exponent, and entropy.

Contribution

It introduces a tropical version of Markov dynamics on the Cayley cubic and establishes its semi-conjugation to a standard torus action, providing new insights into its ergodic behavior.

Findings

The tropical Markov dynamics is semi-conjugated to $SL_2(\mathbb Z)$ action on a torus.

The dynamics is ergodic with Lyapunov exponent given by the spectral radius.

Entropy is computed as the logarithm of the spectral radius.

Abstract

We study the tropical version of Markov dynamics on the Cayley cubic, introduced by V.E. Adler and one of the authors. We show that this action is semi-conjugated to the standard action of $SL_2(\mathbb Z)$ on a torus, and thus is ergodic with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.