题目: On the upper semi-continuity of mertic entropy

报告人: 罗炽逸 ，江西师范大学 

时间: 2025年3月19日（周三）14:40

地点: 东区管理科研楼1418

摘要:

  The upper semi-continuity continuity of the entropy map has attracted significant interest, as it ensures the existence of measures of maximal entropy. A well-known result by Newhouse establishes that forC^∞ diffeomorphisms, the metric entropy is upper semi-continuous. However, for C^r  diffeomorphisms with finite r, this property may fail.

  We prove that for C^(1+) three-dimensional diffeomorphisms, the entropy function is upper semi-continuous under a simple condition: if an invariant measure is a continuity point of the sum of positive Lyapunov exponents, then it is an upper-semi continuity point of the metric entropy. 

  This result not only provides a converse perspective to the work of Buzzi–Crovisier–Sarig (Invent. Math., 2022), which showed that entropy continuity implies Lyapunov exponent continuity for C^∞ surface diffeomorphisms, but also improve some result in Buzzi-Crovisier-Sarig’s paper on SPR property for surface diffeomorphisms.