Dynamical systems workgroup

Invariant measures without Lyapunov exponents

Jorge Olivares

Friday, June 12th, 2020
10:00 AM - 11:00 AM
Zoom meeting

Lyapunov exponent play an important role in the study the ergodic behavior of dynamical systems. In particular the existence and positiveness of Lyapunov exponents has been used since the seminal work of Pesin to study the dynamics of non-uniformly hyperbolic systems. Using this ideas Ledrappier estudied ergodic properties of absolutely continuous invariant measures for $C^2$-maps of the interval under the assumption that the Lyapunov exponent exist and is positive. Recently Doobs develope a non-invertible Pesin theory for interval maps with discontinuities and unbounded derivative as for interval maps with flat critical points and Lima construct a symbolic extension for $C^{1 + \varepsilon}$ maps that code the measures with positive Lyapunov exponent. In this talk we present examples of unimodal maps with Fibonacci recurrence of the orbit of the turning point for which the ergodic measure supported in the $\omega$-limit of this orbit do not have a well-defined Lyapunov exponent and for every point in the post-critical set the pointwise Lyapunov exponent does not exist.

Event contact: vmatusde at ur dot rochester dot edu