Random product of quasi-periodic cocycles

Bezerra, Jamerson; Poletti, Mauricio

doi:10.1090/proc/15428

Random product of quasi-periodic cocycles
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by Jamerson Bezerra and Mauricio Poletti PDF

Proc. Amer. Math. Soc. 149 (2021), 2927-2942 Request permission

Abstract:

Given a finite set of quasi-periodic cocycles the random product of them is defined as the random composition according to some probability measure.

We prove that the set of $C^r$, $0\leq r \leq \infty$ (or analytic) $\kappa +1$-tuples of quasi-periodic cocycles taking values in $SL_{2}(\mathbb {R})$ such that the random product of them has positive Lyapunov exponent contains a $C^0$ open and $C^r$ dense subset which is formed by $C^0$ continuity point of the Lyapunov exponent.

For $\kappa +1$-tuples of quasi-periodic cocycles taking values in $GL_{d}(\mathbb {R})$ for $d>2$, we prove that if one of them is diagonal, then there exists a $C^r$ dense set of such $\kappa +1$-tuples which have simple Lyapunov spectrum and are $C^0$ continuity point of the Lyapunov exponent.

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