Invariant multi-graphs in step skew-products. Issue 1 (2nd January 2020)

• Record Type: Journal Article
• Authors: Gelfert, Katrin; Oliveira, Daniel
• Is Part Of: Dynamical systems. Volume 35:Issue 1(2020)
• Page Start: 1
• Page End: 28
• Publication Date: 2020-01-02
• Subjects: Partial hyperbolicity -- hyperbolic ergodic measures -- skew-product -- invariant graph -- Lyapunov exponents; 37E05 -- 37D30 -- 37D25 -- 37C29; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2019.1616674
• Languages: English
• ISSNs: 1468-9367

Abstract

ABSTRACT: We study step skew-products over a finite-state shift (base) space whose fibre 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.