Properties of Hyperbolic Measures With Local Product Structure
Restricted (Penn State Only)
- Author:
- Alansari, Nawaf
- Graduate Program:
- Mathematics (PHD)
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 27, 2022
- Committee Members:
- Victoria Sadovskaya, Major Field Member
Zhiren Wang, Major Field Member
Yakov Pesin, Chair & Dissertation Advisor
Martin Bojowald, Outside Unit & Field Member
Alexei Novikov, Professor in Charge/Director of Graduate Studies - Keywords:
- Hyperbolic dynamics
ergodic theory
local product structure - Abstract:
- This dissertation studies some properties of hyperbolic measures with local product structure. We first state some known properties of some classical measures with local product structure. Then, for a $C^{1+\alpha}$ diffeomorphism $f$ of a compact smooth manifold, we give a necessary and sufficient condition that guarantees that if the set of Hyperbolic Lyapunov-Perron regular points has positive volume, then $f$ preserves a smooth measure. We use recent results on symbolic coding of $\chi$-non-uniformly hyperbolic sets and results concerning existence of SRB measures for them. We also study ergodic properties of general hyperbolic measures with local product structure. We show that all the classical results that hold in the case of SRB measure hold for these measures. In particular, we show the decomposition in countably many ergodic components, we prove the decomposition into K-components, and show that for hyperbolic measure with local product structure, the K property implies the Bernoulli property. These results are applicable to equilibrium measures, to measures of $u$-maximal entropy for a class of diffeomorphisms, and to a class of local hyperbolic equilibrium measures.