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Invariant multi-graphs in step skew-products

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ABSTRACT We study step skew-products over a finite-state shift (base) space whose fibre maps are injective maps on the unit interval. We show that certain invariant sets have a multi-graph… Click to show full abstract

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

Keywords: multi graphs; step skew; skew products; invariant multi

Journal Title: Dynamical Systems
Year Published: 2019

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