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

Part of: Topological dynamics Low-dimensional dynamical systems

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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References

Brémont, J.. Dynamics of injective quasi-contractions. Ergod. Th. & Dynam. Sys. 26 (2006), 19–44.CrossRefGoogle Scholar
Bugeaud, Y.. Dynamique de certaines applications contractantes, linéaires par morceaux, sur [0, 1[. C. R. Acad. Sci. Sér. 1 Math. 317 (1993), 575–578.Google Scholar
Bugeaud, Y. and Conze, J.-P.. Calcul de la dynamique de transformations linéaires contractantes mod 1 et arbre de Farey. Acta Arith. 88 (1999), 201–218.Google Scholar
Catsigeras, E., Guiraud, P. and Meyroneinc, A.. Complexity of injective piecewise contracting interval maps. Ergod. Th. & Dynam. Sys. 40 (2020), 64–88.CrossRefGoogle Scholar
Catsigeras, E., Guiraud, P., Meyroneinc, A. and Ugalde, E.. On the asymptotic properties of piecewise contracting maps. Dyn. Syst. 31 (2016), 107–135.10.1080/14689367.2015.1068274CrossRefGoogle Scholar
Coutinho, R.. Dinámica simbólica linear. PhD Thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, 1999.Google Scholar
Gambaudo, J. M. and Tresser, C.. On the dynamics of quasi-contractions. Bull. Braz. Math. Soc. (N.S.) 19 (1988), 61–114.CrossRefGoogle Scholar
Laurent, M. and Nogueira, A.. Rotation number of contracted rotations. J. Mod. Dyn. 12 (2018), 175–191.CrossRefGoogle Scholar
Morse, M. and Hedlund, G. A.. Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940), 1–42.CrossRefGoogle Scholar
Nogueira, A. and Pires, B.. Dynamics of piecewise contractions of the interval. Ergod. Th. & Dynam. Sys. 35 (2015), 2198–2215.Google Scholar
Nogueira, A., Pires, B. and Rosales, R. A.. Asymptotically periodic piecewise contractions of the interval. Nonlinearity 27 (2014), 1603–1610.CrossRefGoogle Scholar
Nogueira, A., Pires, B. and Rosales, R. A.. Topological dynamics of piecewise 𝜆-affine maps. Ergod. Th. & Dynam. Sys. 38 (2018), 1876–1893.10.1017/etds.2016.104CrossRefGoogle Scholar
Pires, B.. Symbolic dynamics of piecewise contractions. Nonlinearity 32 (2019), 4871–4889.CrossRefGoogle Scholar