All solenoids of piecewise smooth maps are period doubling

Lluís Alsedà; Víctor Jiménez López; L’ubomír Snoha

Fundamenta Mathematicae (1998)

  • Volume: 157, Issue: 2-3, page 121-138
  • ISSN: 0016-2736

Abstract

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We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if p 1 < . . . < p n is a periodic orbit of a continuous map f then there is a union set q 1 , . . . , q n - 1 of some periodic orbits of f such that p i < q i < p i + 1 for any i.

How to cite

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Alsedà, Lluís, Jiménez López, Víctor, and Snoha, L’ubomír. "All solenoids of piecewise smooth maps are period doubling." Fundamenta Mathematicae 157.2-3 (1998): 121-138. <http://eudml.org/doc/212281>.

@article{Alsedà1998,
abstract = {We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if $p_1 < ... < p_n$ is a periodic orbit of a continuous map f then there is a union set $\{q_1,..., q_\{n-1\}\}$ of some periodic orbits of f such that $p_i < q_i < p_\{i+1\}$ for any i.},
author = {Alsedà, Lluís, Jiménez López, Víctor, Snoha, L’ubomír},
journal = {Fundamenta Mathematicae},
keywords = {Markov graph; periodic point; piecewise smooth map with nowhere vanishing Lipschitz continuous derivative; piecewise linear map; solenoid; piecewise smooth map; period doubling solenoid; periodic orbit},
language = {eng},
number = {2-3},
pages = {121-138},
title = {All solenoids of piecewise smooth maps are period doubling},
url = {http://eudml.org/doc/212281},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Alsedà, Lluís
AU - Jiménez López, Víctor
AU - Snoha, L’ubomír
TI - All solenoids of piecewise smooth maps are period doubling
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 2-3
SP - 121
EP - 138
AB - We show that piecewise smooth maps with a finite number of pieces of monotonicity and nowhere vanishing Lipschitz continuous derivative can have only period doubling solenoids. The proof is based on the fact that if $p_1 < ... < p_n$ is a periodic orbit of a continuous map f then there is a union set ${q_1,..., q_{n-1}}$ of some periodic orbits of f such that $p_i < q_i < p_{i+1}$ for any i.
LA - eng
KW - Markov graph; periodic point; piecewise smooth map with nowhere vanishing Lipschitz continuous derivative; piecewise linear map; solenoid; piecewise smooth map; period doubling solenoid; periodic orbit
UR - http://eudml.org/doc/212281
ER -

References

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  1. [1] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Adv. Ser. in Nonlinear Dynam. 5, World Sci., Singapore, 1993. Zbl0843.58034
  2. [2] A. M. Blokh and M. Yu. Lyubich, Measure and dimension of solenoidal attractors of one dimensional dynamical systems, Comm. Math. Phys. 127 (1990), 573-583. Zbl0721.58033
  3. [3] J. Bobok and M. Kuchta, Register shifts versus transitive F-cycles for piecewise monotone maps, Real Anal. Exchange 21 (1995/96), 134-146. Zbl0849.26002
  4. [4] R. Galeeva and S. van Strien, Which families of l-modal maps are full?, Trans. Amer. Math. Soc. 348 (1996), 3215-3221. Zbl0862.58015
  5. [5] V. Jiménez López and L’. Snoha, There are no piecewise linear maps of type 2 ∞ , ibid. 349 (1997), 1377-1387. Zbl0947.37025
  6. [6] S. F. Kolyada, Interval maps with zero Schwarzian, in: Functional-Differential Equations and Their Applications, Inst. Math. Ukrain. Acad. Sci., Kiev, 1985, 47-57 (in Russian). 
  7. [7] L. Lovász and M. D. Plummer, Matching Theory, Akadémiai Kiadó, Budapest, 1986. 
  8. [8] M. Martens, W. de Melo and S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992), 271-318. Zbl0761.58007
  9. [9] M. Martens and C. Tresser, Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps, Proc. Amer. Math. Soc. 124 (1996), 2863-2870. Zbl0864.58035
  10. [10] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, Berlin, 1993. Zbl0791.58003
  11. [11] M. Misiurewicz, Attracting Cantor set of positive measure for a C ∞ map of an interval, Ergodic Theory Dynam. Systems 2 (1982), 405-415. Zbl0522.58032
  12. [12] C. Preston, Iterates of Piecewise Monotone Mappings on an Interval, Lecture Notes in Math. 1347, Springer, Berlin, 1988. Zbl0684.58002

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