Thermodynamic formalism for contracting Lorenz flows

TL;DR

This paper applies thermodynamic formalism to the contracting Lorenz flow, establishing the existence of equilibrium states and analyzing the Lyapunov spectrum, thereby advancing understanding of its dynamical properties.

Contribution

It proves the existence of equilibrium states for the natural potential in the contracting Lorenz flow and analyzes its Lyapunov spectrum using pressure, a novel application in this context.

Findings

Existence of equilibrium states for the natural potential in the flow.

Analysis of the Lyapunov spectrum via pressure.

Identification of an interval containing t in [0,1] for equilibrium states.

Abstract

We study the expansion properties of the contracting Lorenz flow introduced by Rovella via thermodynamic formalism. Specifically, we prove the existence of an equilibrium state for the natural potential $\hat\phi_t(x,y, z):=-t\log J_{(x, y, z)}^{cu}$ for the contracting Lorenz flow and for $t$ in an interval containing $[0,1]$. We also analyse the Lyapunov spectrum of the flow in terms of the pressure.