Topological conditions for the existence of absorbing Cantor sets

Bruin, Henk

doi:10.1090/S0002-9947-98-02109-6

Topological conditions for the existence of absorbing Cantor sets
HTML articles powered by AMS MathViewer

by Henk Bruin

Trans. Amer. Math. Soc. 350 (1998), 2229-2263

DOI: https://doi.org/10.1090/S0002-9947-98-02109-6

PDF | Request permission

Abstract:

This paper deals with strange attractors of S-unimodal maps $f$. It generalizes earlier results in the sense that very general topological conditions are given that either

[i)] guarantee the existence of an absorbing Cantor set provided the critical point of $f$ is sufficiently degenerate, or
[ii)] prohibit the existence of an absorbing Cantor set altogether.

As a by-product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for $f$.

References

H. Bruin, Topological conditions for the existence of invariant measures for unimodal maps, Ergodic Theory Dynam. Systems 14 (1994), no. 3, 433–451. MR 1293402, DOI 10.1017/S0143385700007963
H. Bruin, Invariant measures of interval maps, Thesis, Delft. (1994).
H. Bruin, Combinatorics of the kneading map, Proceedings of the Conference “Thirty Years after Sharkovskiĭ’s Theorem: New Perspectives” (Murcia, 1994), 1995, pp. 1339–1349. MR 1361922, DOI 10.1142/S0218127495001010
H. Bruin, G. Keller, T. Nowicki, and S. van Strien, Wild Cantor attractors exist, Ann. of Math. (2) 143 (1996), no. 1, 97–130. MR 1370759, DOI 10.2307/2118654
Chuan Xi Wu, A note on stable harmonic mappings, J. Math. (Wuhan) 11 (1991), no. 1, 72–76 (Chinese, with English summary). MR 1139675
A. M. Blokh and M. Yu. Lyubich, Measurable dynamics of $S$-unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 5, 545–573. MR 1132757
John Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), no. 2, 133–160. MR 553966
John Guckenheimer and Stewart Johnson, Distortion of $S$-unimodal maps, Ann. of Math. (2) 132 (1990), no. 1, 71–130. MR 1059936, DOI 10.2307/1971501
Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
Jean-Marc Gambaudo and Charles Tresser, A monotonicity property in one-dimensional dynamics, Symbolic dynamics and its applications (New Haven, CT, 1991) Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 213–222. MR 1185089, DOI 10.1090/conm/135/1185089
Franz Hofbauer, The topological entropy of the transformation $x\mapsto ax(1-x)$, Monatsh. Math. 90 (1980), no. 2, 117–141. MR 595319, DOI 10.1007/BF01303262
Franz Hofbauer and Gerhard Keller, Some remarks on recent results about $S$-unimodal maps, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 4, 413–425. Hyperbolic behaviour of dynamical systems (Paris, 1990). MR 1096100
Stewart D. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys. 110 (1987), no. 2, 185–190. MR 887994
M. Jakobson, G. Świa̧tek, Metric properties of non-renormalizable S-unimodal maps, Preprint IHES/M/91/16 (1991).
Michael Jakobson and Grzegorz Świątek, Metric properties of non-renormalizable $S$-unimodal maps. I. Induced expansion and invariant measures, Ergodic Theory Dynam. Systems 14 (1994), no. 4, 721–755. MR 1304140, DOI 10.1017/S0143385700008130
M. Jakobson, G. Świa̧tek, Quasisymmetric conjugacies between unimodal maps, Preprint Stony Brook 16 (1991).
Gerhard Keller and Tomasz Nowicki, Fibonacci maps re(al)visited, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 99–120. MR 1314971, DOI 10.1017/S0143385700008269
Mikhail Lyubich, Combinatorics, geometry and attractors of quasi-quadratic maps, Ann. of Math. (2) 140 (1994), no. 2, 347–404. MR 1298717, DOI 10.2307/2118604
Mikhail Lyubich, Milnor’s attractors, persistent recurrence and renormalization, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 513–541. MR 1215975
Mikhail Lyubich and John Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2, 425–457. MR 1182670, DOI 10.1090/S0894-0347-1993-1182670-0
M. Martens, Interval Dynamics, Thesis, Delft, 1990.
Marco Martens, Distortion results and invariant Cantor sets of unimodal maps, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 331–349. MR 1279474, DOI 10.1017/S0143385700007902
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
J. Milnor, Local connectivity of Julia sets; Expository lectures, Preprint StonyBrook # 1992/11.
Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
T. Nowicki and S. van Strien, Absolutely continuous invariant measures for $C^2$ unimodal maps satisfying the Collet-Eckmann conditions, Invent. Math. 93 (1988), no. 3, 619–635. MR 952285, DOI 10.1007/BF01410202