Invariant multi-graphs in step skew-products

Abstract

We study step skew-products over a finite-state shift (base) space whose fiber maps are C 1 injective maps on the unit interval. We show that certain invariant sets have a multi-graph structure and can be written graphs of one, two or more functions defined on the base. In particular, this applies to any hyperbolic set and to the support of any ergodic hyperbolic measure. Moreover, within the class of step skew-products whose interval maps are "absorbing", open and densely the phase space decomposes into attracting and repelling double-strips such that their attractors and repellers are graphs of one single-valued or bi-valued continuous function almost everywhere, respectively.

Geometric Structure and Ergodic Properties of Bony Multi-Graphs

“…Definition 2.3.1 (Multi-graph). Following [15], given F ∈ F, a multi-function ψ : D ⊂ Θ → I is a relation that associates to every point θ ∈ D a nonempty subset ψ(θ) ⊂ I. A multi-function ψ : D ⊂ Θ → I is uniformly finite if there exists k ≥ 1 such that #ψ(θ) ≤ k for all θ ∈ D. Given a uniformly finite multi-function ψ, we call the set {(θ, ψ(θ)) : θ ∈ D} a multi-graph in Θ × I.…”

“…In [22], Jäger and Keller provided a criteria, in terms of Lyapunov exponents, for the existence of attracting invariant multi-graphs. Gelfert and Oliveira [15] studied step skew-products over a finite-state shift (base) space whose fiber maps are C 1 injective maps on the unit interval. They provided certain invariant sets having a multi-graph structure and can be written as graphs of one, two or more functions defined on the base.…”

Specification property for step skew products

“…The dynamics of step skew products with interval fibres has been studied by many authors, usually under additional assumptions on fibre maps, see e.g. the recent papers [8,7] and references therein.…”