As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square in (or more generally, of the cube in ) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of (or ). 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.
@article{bwmeta1.element.bwnjournal-article-fmv152i3p267bwm,
author = {H. Bothe},
title = {Shift spaces and attractors in noninvertible horseshoes},
journal = {Fundamenta Mathematicae},
volume = {154},
year = {1997},
pages = {267-289},
zbl = {0884.58063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv152i3p267bwm}
}
Bothe, H. Shift spaces and attractors in noninvertible horseshoes. Fundamenta Mathematicae, Tome 154 (1997) pp. 267-289. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv152i3p267bwm/
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[00004] [5] S. Smale, Diffeomorphisms with many periodic points, in: Differential and Combinatorial Topology, S. S. Cairns (ed.), Princeton Univ. Press, 1963, 63-80.
[00005] [6] C. Tricot, Jr., Two definitions of fractal dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74. | Zbl 0483.28010