# 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

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topAlsedà, 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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- [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] 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
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- [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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