Gradient Flow of the Sinai-Ruelle-Bowen Entropy

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.