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- Title
Gradient Flow of the Sinai–Ruelle–Bowen Entropy.
- Authors
Jiang, Miaohua
- Abstract
Motivated by an extension to Gallavotti–Cohen Chaotic Hypothesis, we study local and global existence of a gradient flow of the Sinai–Ruelle–Bowen entropy functional in the space of transitive Anosov maps. For the space of expanding maps on the unit circle, we equip it with a Hilbert manifold structure using a Sobolev norm in the tangent space of the manifold. Under the additional measure-preserving assumption and a slightly modified metric, we show that the gradient flow exists globally and every trajectory of the flow converges to a unique limiting map where the SRB 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 close connection to a nonlinear partial differential equation, a gradient-dependent diffusion equation.
- Subjects
ORDINARY differential equations; NONLINEAR differential equations; PARTIAL differential equations; TOPOLOGICAL entropy; ENTROPY; HEAT equation; RICCI flow
- Publication
Communications in Mathematical Physics, 2024, Vol 405, Issue 5, p1
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-024-05003-9