Shift spaces and attractors in noninvertible horseshoes

• Autorzy: H. G. Bothe
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

horseshoes; noninvertible maps; shift spaces; attractors

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1997
• Tom: 152
• Numer: 3
• Strony: 267-289

Daty

wydano

1997

otrzymano

1996-04-03

poprawiono

1996-08-01

Twórcy

autor

H. G. Bothe

• Weierstrass-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, D-10117 Berlin, Germany

Bibliografia

• [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.
• [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.