Livsic theorem for matrix cocycles

By B. Kalinin. 2008

141 citations recorded by the source corpus.

We prove the Liv\v{s}ic Theorem for arbitrary $GL(m,\mathbb R)$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a H\"older continuous function $A: X \to GL(m,\mathbb R)$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) ... A(fp) A(p) = Id$ for each periodic point $p=f^n p$, then there exists a H\"older continuous function $C: X \to GL(m,\mathbb R)$ satisfying $A (x) = C(f x) C(x) ^{-1}$ for all $x \in X$. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary H\"older function $A$.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/805b8ebab471a130852cc8ea3e31be70d8d242e4

Citations in the corpus

• Non-conformal Repellers and the Continuity of Pressure for Matrix Cocycles (2013, unknown): “Invariant cones in connection with matrix cocycles and Oseledets’ Theorem were also recently employed by Kalinin in [31], although the setting there is different.”