Gradient Flow of the Sinai-Ruelle-Bowen Entropy

TL;DR

This paper investigates the gradient flow of the Sinai-Ruelle-Bowen entropy on a manifold of expanding circle maps, proving convergence to maximal entropy maps and deriving explicit equations in simple cases.

Contribution

It establishes existence, convergence, and explicit formulas for the gradient flow of the Sinai-Ruelle-Bowen entropy on a Hilbert manifold of expanding maps.

Findings

Gradient flow converges to maps with maximal entropy.

Explicit ODE formula derived in simple cases.

Flow related to a nonlinear diffusion PDE.

Abstract

We study both the local and global existence of a gradient flow of the Sinai-Ruelle-Bowen entropy functional on a Hilbert manifold of expanding maps of a circle equipped with a Sobolev norm in the tangent space of the manifold. We show that, under a slightly modified metric, starting from any initial value, every trajectory of the gradient flow converges to the map with a constant expanding rate where the entropy attains the maximal value. In a simple case, we obtain an explicit formula for the flow's ordinary differential equation representation. This gradient flow has a close connection to a nonlinear partial differential equation: a gradient-dependent diffusion equation.