Real bounds, ergodicity and negative Schwarzian for multimodal maps

van Strien, Sebastian; Vargas, Edson

doi:10.1090/S0894-0347-04-00463-1

Real bounds, ergodicity and negative Schwarzian for multimodal maps
HTML articles powered by AMS MathViewer

by Sebastian van Strien and Edson Vargas

J. Amer. Math. Soc. 17 (2004), 749-782

DOI: https://doi.org/10.1090/S0894-0347-04-00463-1

Published electronically: August 27, 2004

PDF | Request permission

Erratum: J. Amer. Math. Soc. 20 (2007), 267-268.

Abstract:

We consider smooth multimodal maps which have finitely many non-flat critical points. We prove the existence of real bounds. From this we obtain a new proof for the non-existence of wandering intervals, derive extremely useful improved Koebe principles, show that high iterates have ‘negative Schwarzian derivative’ and give results on ergodic properties of the map. One of the main complications in the proofs is that we allow $f$ to have inflection points.

References

A. M. Blokh and M. Yu. Lyubich, Measure and dimension of solenoidal attractors of one-dimensional dynamical systems, Comm. Math. Phys. 127 (1990), no. 3, 573–583. MR 1040895, DOI 10.1007/BF02104502
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, DOI 10.24033/asens.1636
A. M. Blokh and M. Yu. Lyubich, Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. II. The smooth case, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 751–758. MR 1036906, DOI 10.1017/S0143385700005319
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
O. S. Kozlovski, Axiom A maps are dense in the space of unimodal maps in the $C^k$ topology, Ann. of Math. (2) 157 (2003), no. 1, 1–43. MR 1954263, DOI 10.4007/annals.2003.157.1
O. S. Kozlovski, Getting rid of the negative Schwarzian derivative condition, Ann. of Math. (2) 152 (2000), no. 3, 743–762. MR 1815700, DOI 10.2307/2661353

Rigidity for real polynomials

Genadi Levin, Bounds for maps of an interval with one reflecting critical point. I, Fund. Math. 157 (1998), no. 2-3, 287–298. Dedicated to the memory of Wiesław Szlenk. MR 1636895, DOI 10.4064/fm-157-2-3-287-298
Genadi Levin and Sebastian van Strien, Local connectivity of the Julia set of real polynomials, Ann. of Math. (2) 147 (1998), no. 3, 471–541. MR 1637647, DOI 10.2307/120958
Genadi Levin and Sebastian van Strien, Bounds for maps of an interval with one critical point of inflection type. II, Invent. Math. 141 (2000), no. 2, 399–465. MR 1775218, DOI 10.1007/s002220000075
M. Yu. Lyubich, Nonexistence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems. I. The case of negative Schwarzian derivative, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 737–749. MR 1036905, DOI 10.1017/S0143385700005307

Ergodic theory for smooth one-dimensional dynamical systems

Ricardo Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys. 100 (1985), no. 4, 495–524. MR 806250, DOI 10.1007/BF01217727
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
M. Martens, W. de Melo, and S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992), no. 3-4, 273–318. MR 1161268, DOI 10.1007/BF02392981
W. de Melo and S. van Strien, A structure theorem in one-dimensional dynamics, Ann. of Math. (2) 129 (1989), no. 3, 519–546. MR 997312, DOI 10.2307/1971516
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
Weixiao Shen, Bounds for one-dimensional maps without inflection critical points, J. Math. Sci. Univ. Tokyo 10 (2003), no. 1, 41–88. MR 1963798
Weixiao Shen, On the measurable dynamics of real rational functions, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 957–983. MR 1992673, DOI 10.1017/S0143385702001311
Weixiao Shen, On the metric properties of multimodal interval maps and $C^2$ density of Axiom A, Invent. Math. 156 (2004), no. 2, 301–403. MR 2052610, DOI 10.1007/s00222-003-0343-2
Grzegorz Świa̧tek and Edson Vargas, Decay of geometry in the cubic family, Ergodic Theory Dynam. Systems 18 (1998), no. 5, 1311–1329. MR 1653256, DOI 10.1017/S0143385798117558
Sebastian van Strien, Hyperbolicity and invariant measures for general $C^2$ interval maps satisfying the Misiurewicz condition, Comm. Math. Phys. 128 (1990), no. 3, 437–495. MR 1045879, DOI 10.1007/BF02096868
E. Vargas, Measure of minimal sets of polymodal maps, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 159–178. MR 1375131, DOI 10.1017/S0143385700008750