Thematic Program on
Dynamical Systems
 School of Mathematics, IPM,
 February - May, 2017

School of Mathematics
 & Topology

Practical Information  

Mini Course

Alejandro Kocsard
(Universidade Federal Fluminense, Brazil)

Title: Cocycles over hyperbolic dynamical systems, group actions and random walks on manifolds

Date &
Tuesday, May 16, 2017, 15:00--16:15
Tuesday, May 16, 2017, 16:45--18:00

Thursday, May 18, 2017, 15:00--16:15
Thursday, May 18, 2017, 16:45--18:00

Lecture Hall 2,
IPM Niavaran Building,
Niavaran Square, Tehran

In hyperbolic dynamics there is a general (and vague) idea that can be summarized saying with the following sentence:
Most of the dynamically interesting information on a hyperbolic system is concentrated in its periodic orbits.
In this mini-course we shall mainly deal with cocycles over hyperbolic dynamical systems and discuss some results supporting this idea (like Livšic type theorems) and some others contradicting it. Finally, we shall discuss some applications of these techniques to the study of finitely generated groups of diffeomorphisms and random walks on manifolds.

The fundamental references for the mini-course are the following: [6], [3], [1], [4], [5] ,[2].

[1]  A. Avila and M. Viana, Extremal Lyapunov exponents: an invariance principle and applications, Invent. Math. 181 (2010), no. 1, 115–189.

[2]  L. Backes and A. Kocsard, Cohomology of dominated diffeomorphism-valued cocycles over hyperbolic systems, Ergodic Theory Dynam. Systems 36 (2016), 1703–1722.

[3]  B. Hasselblatt, Hyperbolic dynamical systems, Handbook of Dynamical Systems, vol. 1, Elsevier, 2002, pp. 239–319.

[4]  B. Kalinin,  Livšic theorem for matrix cocycles, Ann. of Math. 173 (2011), no. 2, 1025–1042.

[5]  A. Kocsard and R. Potrie, Livšic theorem for low-dimensional diffeomorphism cocycles, Comment. Math. Helv. 91 (2016), 39–64.

[6]  A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320.

School of Mathematics,
IPM - Institute for Research in Fundamental Sciences
Niavaran Building, Niavaran Square, Tehran, Iran
Tel: +98 21 222 90 928