Genericity of historic behavior for maps and flows
Abstract
We establish a sufficient condition for a continuous map, acting on a compact metric space, to have a Baire residual set of points exhibiting historic behavior (also known as irregular points). This criterion applies, for instance, to a minimal and non-uniquely ergodic map; to maps preserving two distinct probability measures with full support; to non-trivial homoclinic classes; to some non-uniformly expanding maps; and to partially hyperbolic diffeomorphisms with two periodic points whose stable manifolds are dense, including Mañé and Shub examples of robustly transitive diffeomorphisms. This way, our unifying approach recovers a collection of known deep theorems on the genericity of the irregular set, for both additive and sub-additive potentials, and also provides a number of new applications.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2021
- DOI:
- 10.48550/arXiv.2107.01200
- arXiv:
- arXiv:2107.01200
- Bibcode:
- 2021arXiv210701200C
- Keywords:
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- Mathematics - Dynamical Systems
- E-Print:
- 14 pages, revised and improved version of previous preprint "Minimality and irregular sets"