Kyoto Dynamics Day 11

日時 (Date) :

2013年2月18日(月) (18 February, 2013)

会場 (Place) :

京都大学 大学院理学研究科 理学部３号館108講演室 (Kyoto Univ., Building No. 3, Room 108)
〒606-8502 京都市左京区北白川追分町, 市バス京大農学部前下車
アクセス/ Access

講演者 (Speakers) :

Lorenzo J. Diaz 氏（PUC-Rio)

篠原 克寿 氏 (Katsutoshi Shinohara, 京都大学，Universite de Bourgogne)

プログラム PROGRAM

10:30-11:30 Diaz (1)

Robust vanishing of all central Lyapunov exponents

(11:30-13:00 Lunch)

13:00-14:00 Shinohara (1)

A C^1-generic failure of Pesin theory, Part I

14:20-15:20 Diaz (2)

Porcupine-like horsehoes: transitivity, Lyapunov spectrum, and phase transitions

15:40-16:40 Shinohara (2)

A C^1-generic failure of Pesin theory, Part II

(18:00- Dinner)

18日の18時より懇親会を行います．当日の昼までに懇親会参加者の受付をするので， 午後から研究集会に参加される方は事前に浅岡まで連絡をお願いします．

アブストラクト Abstracts

Lorenzo J. Diaz 氏 :

(1) "Robust vanishing of all central Lyapunov exponents"

In the skew product and/or partially hyperbolic settings, we will discuss the construction of ergodic measures with a zero Lyapunov exponent (non-hyperbolic measures) explaining the difficulty of constructing measures with several zero exponents. We next describe how to bypass this problem constructing open sets of iterated function systems on arbitrary compact manifolds admitting fully supported ergodic measures all whose Lyapunov exponents vanish. We discuss the differences in the C1 and C2 cases. We finally exploit the consequences for partially hyperbolic maps. This is a join work with J. Bochi and Ch. Bonatti.

(2) "Porcupine-like horsehoes: transitivity, Lyapunov spectrum, and phase transitions"

We discuss simple, but representative, examples of local diffeomorphisms defined as one-step skew products modeled over a horsehose map. These systems are naturally asociated to a heterodimensional cylce. This cycle gives rise to a homoclinic class on which the diffeomorphism is topologically transitive and partially hyperbolic. It can be conveniently studied in terms of an iterated function system of interval maps that are genuinely non-contracting. These examples have topologically a rich fiber structure (justifying the porcupine terminology). Moreover, they exhibit a rich phase transition in the pressure function (coexistence of equilibrium states with positive entropies) that is associated to a gap in the spectrum of Lyapunov exponents in the central direction. This is a joint work with K. Gelfert (UFRJ) and M. Rams (IM PAN Warsaw).

篠原克寿氏 (Katsutoshi Shinohara) :

"A C^1-generic failure of Pesin theory (part 1 and 2)"

Given a $C^{1+\alpha}$ diffeomorphism and its invariant ergodic measure, if it is hyperbolic (i.e., all the Lyapnov exponents are non-zero) then Pesin theory tells us the existence of the stable and the unstable manifolds at almost everywhere compatible with Oceledet's splitting. Because of its wide spectrum of applications, it has been a strong tool for the research of non-uniformly hyperbolic dynamics. One may wonder, for the deeper understanding of original Pesin theory, the application to the study of $C^1$-generic dynamics, or just for fun, if it is still valid for diffeomorphisms which is only $C^1$. The aim of this talk is to show that there does exist a region of $C^1$-diffeomorphisms where Pesin theory fails, and such a failure can be observed very frequently there. My talk will be split into two parts: in the first part, I will give the axiomatic description of our examples and explain that they really are our examples. In the second part, I will explain how we construct such examples. Main ingredient is the $C^1$-perturbation techniques developed by Bonatti, Diaz, Crovisier and Gourmelon.