Shift spaces and attractors in noninvertible horseshoes

H. Bothe

Fundamenta Mathematicae (1997)

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

Abstract

top
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.

How to cite

top

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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.