Continuity properties of Lyapunov exponents for surface diffeomorphisms

• 成果类型: Article
• 署名作者: Buzzi, Jerome; Crovisier, Sylvain; Sarig, Omri
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Weizmann Institute of Science
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01132-x
• 发表日期: 2022
• 页码: 767-849
• 关键词: symbolic extensions; Metric Entropy; volume growth; dimension; escape; mass

摘要

We study the entropy and Lyapunov exponents of invariant measures mu for smooth surface diffeomorphisms f, as functions of (f, mu). The main result is an inequality relating the discontinuities of these functions. One consequence is that for a C-infinity surface diffeomorphism, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semicontinuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy.