On the Lyapunov spectrum of relative transfer operators

Abstract

We analyze the Lyapunov spectrum of the relative Ruelle operator associated with a skew product whose base is an ergodic automorphism and whose fibers are full shifts. We prove that these operators can be approximated in the $C^0$-topology by positive matrices with an associated dominated splitting.Comment: The article now contains a section on decay of correlations of relative transfer operator

Continuity of Lyapunov exponents is equivalent to continuity of Oseledets subspaces

“…While the first option is quite expected since dominated splittings vary continuously, the second one is not so evident. Moreover, using results of [BeS16] we get a similar conclusion in the semi-invertible setting when restricted to stochastic matrices.…”

Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism

“…The proof of this claim follows from similar arguments as the ones appeared in Section 3 in [9] and [11], which in turn are inspired in the references [7,22]. Basically these authors proved that the dual of an auxiliary Markov operator constructed from L A contracts the Wasserstein distance on M(l p (R)).…”

Thermodynamic formalism for topological Markov chains on standard Borel spaces

“…Observe that, by construction, P m n (1) = 1 and P m k+l • P k l = P k+m l . Furthermore, the proof of Lemma 2.1 in [BS16] is also applicable to the situation in here and gives that…”

“…Since P m n contracts d, it immediately follows from the composition rule that, for any probability measure ν 0 ∈ M 1 (X), the sequence ((P m 0 ) * (ν 0 )) m∈N is a Cauchy sequence and therefore converges to a probability measure ν, which, again by contraction, is independent from ν 0 . It then follows as in [BS16] that ν is a conformal measure, that is, L * f (ν) = λν for some λ > 0. Observe that, by conformality, λ = L (1)dν.…”