Entropy along expanding foliations
By Jiagang Yang. 2016
74 citations recorded by the source corpus.
The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate of complexity generated by the diffeomorphism along the leaves of the foliation. We prove that this number varies upper semi-continuously with the diffeomorphism ($\C^1$ topology), the invariant measure (weak* topology) and the foliation itself in a suitable sense. This has several important consequences. For one thing, it implies that the set of Gibbs $u$-states of $\C^{1+}$ partially hyperbolic diffeomorphisms is an upper semi-continuous function of the map in the $\C^1$ topology. Another consequence is that the sets of partially hyperbolic diffeomorphisms with mostly contracting or mostly expanding center are $\C^1$ open. New examples of partially hyperbolic diffeomorphisms with mostly expanding center are provided, and the existence of physical measures for $C^1$ residual subset of diffeomorphisms are discussed. We also provide a new class of robustly transitive diffeomorphisms: every $C^2$ volume preserving, accessible partially hyperbolic diffeomorphism with one dimensional center and non-vanishing center exponent is $C^1$ robustly transitive (among neighborhood of diffeomorphisms which are not necessarily volume preserving).
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/def0af5cf977d2dbbab22a674f67b1d42a3f3678
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In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Unstable Entropies and Variational Principle for Partially Hyperbolic Diffeomorphisms (2017, methodology): “on subordinate to unstable manifolds. Because the partition is increasing, it is convenient to use sometimes. However, practically it takes some work to construct such partitions, such as in [12] and [19]. In our definition, instead of ξ, we relax the increasing condition and use partitions η that ca”