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A spectral decomposition of the attractor of piecewise-contracting maps of the interval

Part of: Low-dimensional dynamical systems Topological dynamics

Published online by Cambridge University Press:  05 May 2020

ALFREDO CALDERON Open the ORCID record for ALFREDO CALDERON [Opens in a new window] ,
ELEONORA CATSIGERAS Open the ORCID record for ELEONORA CATSIGERAS [Opens in a new window]  and
PIERRE GUIRAUD Open the ORCID record for PIERRE GUIRAUD [Opens in a new window]
ALFREDO CALDERON
Affiliation:
Instituto de Ingeniería Matemática and Centro de Investigación y Modelamiento de Fenómenos Aleatorios Valparaíso, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso, Chile email alfredo.calderon@postgrado.uv.cl, pierre.guiraud@uv.cl
ELEONORA CATSIGERAS
Affiliation:
Instituto de Matemática y Estadística Rafael Laguardia, Universidad de la República, Montevideo, Uruguay email eleonora@fing.edu.uy
PIERRE GUIRAUD
Affiliation:
Instituto de Ingeniería Matemática and Centro de Investigación y Modelamiento de Fenómenos Aleatorios Valparaíso, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso, Chile email alfredo.calderon@postgrado.uv.cl, pierre.guiraud@uv.cl
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Abstract

We study the asymptotic dynamics of piecewise-contracting maps defined on a compact interval. For maps that are not necessarily injective, but have a finite number of local extrema and discontinuity points, we prove the existence of a decomposition of the support of the asymptotic dynamics into a finite number of minimal components. Each component is either a periodic orbit or a minimal Cantor set and such that the $\unicode[STIX]{x1D714}$-limit set of (almost) every point in the interval is exactly one of these components. Moreover, we show that each component is the $\unicode[STIX]{x1D714}$-limit set, or the closure of the orbit, of a one-sided limit of the map at a discontinuity point or at a local extremum.

Keywords

interval mappiecewise contractionperiodic attractorminimal Cantor set

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37B20: Notions of recurrence
Secondary: 37B35: Gradient-like and recurrent behavior; isolated (locally maximal) invariant sets
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 1940 - 1960
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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