The entropy of Lyapunov-optimizing measures for some matrix cocycles

Abstract: Consider a simple (to formulate...) mathematical object: you are given a finite collection of matrices $A_i\in GL(2,\mathbb R); i=1,\ldots,k$ and you are allowed to multiply them, in any order. The notion you are interested in is the exponential rate of speed of growth of the norm: given $\omega\in \{1,\ldots,k\}^{\mathbb N}$, let \[ \lambda(\omega) = \lim_{n\to\infty} \frac 1n \log ||A_{\omega_n} \cdot \ldots \cdot A_{\omega_1}||. \] This object has many names, in dynamical systems we call it the Lyapunov exponent.

We are in particular interested in the set of those $\omega$'s that give the extremal (maximal, minimal) value of the Lyapunov exponent. A long-standing conjecture states that for a generic matrix collection those sets ought to be {\it small}, in some sense. In the result I will present we (Jairo Bochi and me) are proving that for certain open set of collections of matrices those $\omega$'s that maximize/minimize Lyapunov exponent have zero topological entropy.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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