Hostname: page-component-7dd5485656-zklqj Total loading time: 0 Render date: 2025-10-31T14:05:57.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Multi-dimensional piecewise contractions are asymptotically periodic

Part of: Smooth dynamical systems: general theory

Published online by Cambridge University Press:  30 October 2025

JOSÉ PEDRO GAIVÃO  and
BENITO PIRES
JOSÉ PEDRO GAIVÃO
Affiliation:
Departamento de Matemática, Instituto Superior de Economia e Gestão da Universidade de Lisboa , Portugal e-mail: jpgaivao@iseg.ulisboa.pt
BENITO PIRES*
Affiliation:
Departamento de Computação e Matemática, Faculdade de Filosofia, Ciâncias e Letras da Universidade de São Paulo , Brazil
*
e-mail: benito@usp.br
Get access Rights & Permissions [Opens in a new window]

Abstract

Piecewise contractions (PCs) are piecewise smooth maps that decrease the distance between pairs of points in the same domain of continuity. The dynamics of a variety of systems is described by PCs. During the last decade, much effort has been devoted to proving that in parameterized families of one-dimensional PCs, the $\omega $-limit set of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor $\omega $-limit sets. In this article, we extend these results to the multi-dimensional case. More precisely, we provide criteria to show that an arbitrary family $\{f_{\mu }\}_{\mu \in U}$ of locally bi-Lipschitz piecewise contractions $f_\mu :X\to X$ defined on a compact metric space X is asymptotically periodic for Lebesgue almost every parameter $\mu $ running over an open subset U of the M-dimensional Euclidean space $\mathbb {R}^M$. As a corollary of our results, we prove that piecewise affine contractions of $\mathbb {R}^d$ defined in generic polyhedral partitions are asymptotically periodic.

Keywords

piecewise contractionperiodic orbitsperiodic attractor

MSC classification

Secondary: 37C25: Fixed points, periodic points, fixed-point index theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 26
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Antunes, A. A., Bugeaud, Y. and Pires, B.. Switched server systems whose parameters are normal numbers in base 4. Qual. Theory Dyn. Syst. 21(143) (2022), 1–13.CrossRefGoogle Scholar
Bergthaller, C. and Singer, I.. The distance to a polyhedron. Linear Algebra Appl. 169 (1992), 111–129.10.1016/0024-3795(92)90174-9CrossRefGoogle Scholar
Blank, M. and Bunimovich, L.. Switched flow systems: pseudo billiard dynamics. Dyn. Syst. 19(4) (2004), 359–370.CrossRefGoogle Scholar
Bowman, J. P. and Sanderson, S.. Angels’ staircases, Sturmian sequences, and trajectories on homothety surfaces. J. Mod. Dyn. 16 (2020), 109–153.CrossRefGoogle Scholar
Bruin, H. and Deane, J. H. B.. Piecewise contractions are asymptotically periodic. Proc. Amer. Math. Soc. 137(4) (2008), 1389–1395.10.1090/S0002-9939-08-09633-0CrossRefGoogle Scholar
Bugeaud, Y.. Dynamique de certaines applications contractantes, linéaires par morceaux, sur $\left[0,1\right)$ . C. R. Acad. Sci. Paris Sér. I Math. 317(6) (1993), 575–578.Google Scholar
Bugeaud, Y.. Linear mod one transformations and the distribution of fractional parts $\{\xi {(p/q)}^n\}$ . Acta Arith. 114(4) (2004), 301–311.CrossRefGoogle Scholar
Bugeaud, Y., Kim, D. H., Laurent, M. and Nogueira, A.. On the Diophantine nature of the elements of Cantor sets arising in the dynamics of contracted rotations. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22(4) (2021), 1691–1704.Google Scholar
Buzzi, J.. Intrinsic ergodicity of affine maps in ${\left[0,1\right]}^d$ . Monatsh. Math. 124(2) (1997), 97–118.CrossRefGoogle Scholar
Calderon, A., Catsigeras, E. and Guiraud, P.. A spectral decomposition of the attractor of piecewise-contracting maps of the interval. Ergod. Th. & Dynam. Sys. 41(7) (2021), 1940–1960.CrossRefGoogle Scholar
Catsigeras, E. and Guiraud, P.. Integrate and fire neural networks, piecewise contractive maps and limit cycles. J. Math. Biol. 67(3) (2013), 609–655.CrossRefGoogle ScholarPubMed
Catsigeras, E., Guiraud, P. and Meyroneinc, A.. Complexity of injective piecewise contracting interval maps. Ergod. Th. & Dynam. Sys. 40(1) (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(2) (2016), 107–135.CrossRefGoogle Scholar
Chase, C., Serrano, J. and Ramadge, P. J.. Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous systems. IEEE Trans. Automat. Control 38(1) (1993), 70–83.CrossRefGoogle Scholar
Del Magno, G., Gaivão, J. P. and Gutkin, E.. Dissipative outer billiards: a case study. Dyn. Syst. 30(11) (2015), 45–69.10.1080/14689367.2014.970130CrossRefGoogle Scholar
Fernandes, F. and Pires, B.. A switched server system semiconjugate to a minimal interval exchange. European J. Appl. Math. 31(4) (2020), 682–708.10.1017/S095679251900024XCrossRefGoogle Scholar
Gaivão, J. P.. Asymptotic periodicity in outer billiards with contraction. Ergod. Th. & Dynam. Sys. 40(2) (2020), 402–417.CrossRefGoogle Scholar
Gaivão, J. P.. Hausdorff dimension of the exceptional set of interval piecewise affine contractions. Qual. Theory Dyn. Syst. 24 (2025), 86.CrossRefGoogle Scholar
Gaivão, J. P., Laurent, M. and Nogueira, A.. Rotation number of 2-interval piecewise affine maps. Aequ. Math. 99(2) (2025), 511–530.CrossRefGoogle Scholar
Gaivão, J. P. and Nogueira, A.. Dynamics of piecewise increasing contractions. Bull. Lond. Math. Soc. 54(2) (2022), 482–500.10.1112/blms.12577CrossRefGoogle Scholar
Gaivão, J. P. and Peixe, T.. Periodic attractor in the discrete time best-response dynamics of the rock-paper-scissors game. Dyn. Games Appl. 11(3) (2021), 491–511.CrossRefGoogle Scholar
Gutierrez, C.. Smoothability of Cherry flows on two-manifolds. Lect. Notes Math. 1007 (1983), 308–331.10.1007/BFb0061422CrossRefGoogle Scholar
Gutierrez, C. and Pires, B.. On ${C}^r$ -closing for flows on orientable and non-orientable 2-manifolds. Bull. Braz. Math. Soc. (N.S.) 40(4) (2009), 553–576.CrossRefGoogle Scholar
Jain, S. and Liverani, C.. Piecewise contractions. Ergod. Th. & Dynam. Sys. 45(5) (2025), 1503–1540.CrossRefGoogle Scholar
Janson, S. and Öberg, A.. A piecewise contractive dynamical system and Phragmén’s election method. Bull. Soc. Math. France 147(3) (2019), 395–441.CrossRefGoogle Scholar
Jeong, I.-J.. Outer billiards with contraction: attracting Cantor sets. Exp. Math. 24(1) (2015), 53–64.CrossRefGoogle Scholar
Jeong, I.-J.. Outer billiards with contraction: regular polygons. Dyn. Syst. 33(4) (2018), 565–580.CrossRefGoogle Scholar
Kruglikov, B. and Rypdal, M.. A piece-wise affine contracting map with positive entropy. Discrete Contin. Dyn. Syst. 16(2) (2006), 393–394.CrossRefGoogle Scholar
Laurent, M. and Nogueira, A.. Rotation number of contracted rotations. J. Mod. Dyn. 12 (2018), 175–191.10.3934/jmd.2018007CrossRefGoogle Scholar
Laurent, M. and Nogueira, A.. Dynamics of 2-interval piecewise affine maps and Hecke–Mahler series. J. Mod. Dyn. 17 (2021), 33–63.10.3934/jmd.2021002CrossRefGoogle Scholar
MacPhee, I. M., Menshikov, M. V., Popov, S. and Volkov, S.. Periodicity in the transient regime of exhaustive polling systems. Ann. Appl. Probab. 16(4) (2006), 1816–1850.CrossRefGoogle Scholar
Nogueira, A., Pires, B. and Rosales, R. A.. Asymptotically periodic piecewise contractions of the interval. Nonlinearity 27(7) (2014), 1603–1610.CrossRefGoogle Scholar
Nogueira, A., Pires, B. and Rosales, R. A.. Topological dynamics of piecewise $\lambda$ -affine maps. Ergod. Th. & Dynam. Sys. 38(5) (2018), 1876–1893.CrossRefGoogle Scholar
Pires, B.. Symbolic dynamics of piecewise contractions. Nonlinearity 32(12) (2019), 4871–4889.CrossRefGoogle Scholar
Pires, B.. Piecewise contractions and $b$ -adic expansions. Math. Rep. Sci. Canada 42(1) (2020), 1–9.Google Scholar
Pires, B.. A chaotic discrete-time continuous-state Hopfield network with piecewise-affine activation functions. Discrete Contin. Dyn. Syst. Ser. B 28(9) (2023), 4683–4691.CrossRefGoogle Scholar
Winder, R. O.. Partitions of $N$ -space by hyperplanes. SIAM J. Appl. Math. 14 (1966), 811–818.10.1137/0114068CrossRefGoogle Scholar