Skip to main content

Complexity of injective piecewise contracting interval maps

  • E. CATSIGERAS (a1), P. GUIRAUD (a2) and A. MEYRONEINC (a3)

We study the complexity of the itineraries of injective piecewise contracting maps on the interval. We prove that for any such map the complexity function of any itinerary is eventually affine. We also prove that the growth rate of the complexity is bounded from above by the number, $N-1$ , of discontinuities of the map. To show that this bound is optimal, we construct piecewise affine contracting maps whose itineraries all have the complexity $(N-1)n+1$ . In these examples, the asymptotic dynamics take place in a minimal Cantor set containing all the discontinuities.

Hide All
[1] Alessandri, P. and Berthé, V.. Three distance theorems and combinatorics on words. Enseign. Math. 44 (1998), 103132.
[2] Brémont, J.. Dynamics of injective quasi-contractions. Ergod. Th. & Dynam. Sys. 26 (2006), 1944.
[3] Bugeaud, Y.. Dynamique de certaines applications contractantes linéaires par morceaux sur [0, 1[. C. R. Math. Acad. Sci. Paris, I 317 (1993), 575578.
[4] Bugeaud, Y. and Conze, J.-P.. Calcul de la dynamique de transformations linéaires contractantes mod 1 et arbre de Farey. Acta Arith. 88 (1999), 201218.
[5] Catsigeras, E., Guiraud, P., Meyroneinc, A. and Ugalde, E.. On the asymptotic properties of piecewise contracting maps. Dyn. Syst. 67 (2016), 609655.
[6] Coutinho, R.. Dinâmica simbólica linear. PhD Thesis, Technical University of Lisbon, 1999.
[7] Coutinho, R., Fernandez, B., Lima, R. and Meyroneinc, A.. Discrete time piecewise affine models of genetic regulatory networks. J. Math. Biol. 52 (2006), 524570.
[8] Didier, G.. Caractérisation des N-écritures et application à l’étude des suites de complexité ultimement n + cste . Theoret. Comput. Sci. 215 (1999), 3149.
[9] Gambaudo, J.-M. and Tresser, Ch.. On the dynamics of quasi-contractions. Bull. Braz. Math. Soc. (N.S.) 19 (1988), 61114.
[10] Kruglikov, B. and Rypdal, M.. A piecewise affine contracting map with positive entropy. Discrete Contin. Dyn. Syst. 16 (2006), 393394.
[11] Lima, R. and Ugalde, E.. Dynamical complexity of discrete-time regulatory networks. Nonlinearity 19 (2006), 237259.
[12] Nogueira, A., Pires, B. and Rosales, R.. Asymptotically periodic piecewise contractions of the interval. Nonlinearity 27 (2014), 16031610.
[13] Nogueira, A. and Pires, B.. Dynamics of piecewise contractions of the interval. Ergod. Th. & Dynam. Sys. 35 (2015), 21982215.
[14] Nogueira, A., Pires, B. and Rosales, R.. Topological dynamics of piecewise 𝜆-affine maps. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2016.104. Published online 10 November 2016.
[15] Pires, B.. Invariant measures for piecewise continuous maps. C. R. Math. Acad. Sci. Paris, I 354 (2016), 717722.
[16] Veerman, P.. Symbolic dynamics of order-preserving orbits. Physica D 29 (1987), 191201.
[17] Ferenczi, S.. Combinatorial methods for interval exchange transformations. Southeast Asian Bull. Math. 37 (2013), 4766.
[18] Ferenczi, S. and Zamboni, C.. Languages of k-interval exchange transformations. Bull. Lond. Math. Soc. 40 (2008), 705714.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed