# Shift spaces and attractors in noninvertible horseshoes

Fundamenta Mathematicae (1997)

- Volume: 152, Issue: 3, page 267-289
- ISSN: 0016-2736

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topBothe, H.. "Shift spaces and attractors in noninvertible horseshoes." Fundamenta Mathematicae 152.3 (1997): 267-289. <http://eudml.org/doc/212211>.

@article{Bothe1997,

abstract = {As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square $I^2$ in $ℝ^2$ (or more generally, of the cube $I^m$ in $ℝ^m$) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of $I^2$ (or $I^m$). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.},

author = {Bothe, H.},

journal = {Fundamenta Mathematicae},

keywords = {horseshoes; noninvertible maps; shift spaces; attractors},

language = {eng},

number = {3},

pages = {267-289},

title = {Shift spaces and attractors in noninvertible horseshoes},

url = {http://eudml.org/doc/212211},

volume = {152},

year = {1997},

}

TY - JOUR

AU - Bothe, H.

TI - Shift spaces and attractors in noninvertible horseshoes

JO - Fundamenta Mathematicae

PY - 1997

VL - 152

IS - 3

SP - 267

EP - 289

AB - As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square $I^2$ in $ℝ^2$ (or more generally, of the cube $I^m$ in $ℝ^m$) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of $I^2$ (or $I^m$). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.

LA - eng

KW - horseshoes; noninvertible maps; shift spaces; attractors

UR - http://eudml.org/doc/212211

ER -

## References

top- [1] H. G. Bothe, Attractors of noninvertible maps, Preprint 77, IAAS, 1993.
- [2] K. Falconer, Fractal Geometry, Wiley, 1990.
- [3] J. Moser, Stable and Random Motions in Dynamical Systems, Ann. of Math. Stud. 77, Princeton Univ. Press, 1973. Zbl0271.70009
- [4] F. Przytycki, On Ω-stability and structural stability of endomorphisms satisfying Axiom A, Studia Math. 60 (1977), 61-77.
- [5] S. Smale, Diffeomorphisms with many periodic points, in: Differential and Combinatorial Topology, S. S. Cairns (ed.), Princeton Univ. Press, 1963, 63-80.
- [6] C. Tricot, Jr., Two definitions of fractal dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74. Zbl0483.28010

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