第19回DCSセミナー Thermodynamics of towers and the liftability problem 他

開催日時
2012年   3月 12日 15時 00分 ~  
場所
北海道大学 理学部3号館 3-205
講演者
Yakov Pesin (Pennsylvania State Univ.)
 
a) Title: "Thermodynamics of towers and the liftability problem"

Abstract: Thermodynamic formalism of statistical physics is a collection of results aimed at producing some "natural" invariant measures with strong ergodic properties including Sinai-Ruelle-Bowen measures (absolutely continuous invariant measures in the one-dimensional case), measures of maximal dimension and entropy. In particular, one can study the pressure function, which plays crucial role in multifractal analysis and in applications to number theory. In the classical situations (Anosov maps, hyperbolic attractors, etc.) thermodynamical formalism can be effected using symbolic representations of these systems via subshifts of finite type. Many non-classical situations (e.g., one-dimensional unimodal maps and Henon attractors) can be handled using symbolic representations via towers whose base is the Bernoulli shift on a countable set of states. I will describe general tower constructions and present some recent results on thermodynamics of the corresponding systems and in particular, on the study of the pressure function. The principle new phenomenon is that one may have to reduce the class of invariant measures under consideration to the so-called liftable measures. I will describe these measures and discuss the associated liftability problem.

b) Title: "A general approach for constructing SRB-measures for chaotic attractors"

Abstract: I discuss a general approach to the existence of SRB measures for local diffeomorphisms possessing chaotic attractors. I introduce a certain recurrence condition on the iterates of Lebesgue measure called “usable hyperbolicity” that is a version of Lyapunov?Perron regularity in the dissipative case. I will show that if the asymptotic rate of usable hyperbolicity is non-zero on a set of positive Lebesgue measure, then the system has an SRB measure. Some examples will be discussed.

DCS_poster_March12

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