Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-29T09:47:49.665Z Has data issue: false hasContentIssue false

Invariant measures for interval maps without Lyapunov exponents

Published online by Cambridge University Press:  15 November 2021

JORGE OLIVARES-VINALES*
Affiliation:
Department of Mathematics, University of Rochester, Hylan Building, Rochester, NY 14627, USA Departamento de Ingeniería Matemática, Universidad de Chile, Beauchef 851, Santiago, Chile

Abstract

We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.

Type
Original Article
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Araújo, V., Luzzatto, S. and Viana, M.. Invariant measures for interval maps with critical points and singularities. Adv. Math. 221(5) (2009), 14281444.CrossRefGoogle Scholar
Branner, B. and Hubbard, J. H.. The iteration of cubic polynomials. II. Patterns and parapatterns. Acta Math. 169(3–4) (1992), 229325.CrossRefGoogle Scholar
Bruin, H., Keller, G., Nowicki, T. and van Strien, S.. Wild Cantor attractors exist. Ann. of Math. (2) 143(1) (1996), 97130.CrossRefGoogle Scholar
Bruin, H.. Minimal Cantor systems and unimodal maps. J. Difference Equ. Appl. 9 (2003), 305318. Dedicated to Professor Alexander N. Sharkovsky on the occasion of his 65th birthday.CrossRefGoogle Scholar
Collet, P. and Eckmann, J.-P.. Positive Liapunov exponents and absolute continuity for maps of the interval. Ergod. Th. & Dynam. Sys. 3(1) (1983), 1346.CrossRefGoogle Scholar
Cortez, M. I. and Rivera-Letelier, J.. Invariant measures of minimal post-critical sets of logistic maps. Israel J. Math. 176 (2010), 157193.CrossRefGoogle Scholar
de Melo, W. and van Strien, S.. One-Dimensional Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 25). Springer-Verlag, Berlin, 1993.CrossRefGoogle Scholar
Dobbs, N.. On cusps and flat tops. Ann. Inst. Fourier (Grenoble) 64(2) (2014), 571605.CrossRefGoogle Scholar
Dobbs, N.. Pesin theory and equilibrium measures on the interval. Fund. Math. 231(1) (2015), 117.CrossRefGoogle Scholar
Denker, M., Przytycki, F. and Urbański, M.. On the transfer operator for rational functions on the Riemann sphere. Ergod. Th. & Dynam. Sys. 16(2) (1996), 255266.CrossRefGoogle Scholar
Gao, B. and Shen, W.. Summability implies Collet–Eckmann almost surely. Ergod. Th. & Dynam. Sys. 34(4) (2014), 11841209.Google Scholar
Gao, R. and Shen, W. X.. Decay of correlations for Fibonacci unimodal interval maps. Acta Math. Sin. (Engl. Ser.) 34(1) (2018), 114138.Google Scholar
Guckenheimer, J. and Williams, R. F.. Structural stability of Lorenz attractors. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 5972.CrossRefGoogle Scholar
Hofbauer, F. and Keller, G.. Some remarks on recent results about $S$ -unimodal maps. Ann. Inst. Henri Poincaré Phys. Théor. 53 (1990), 413425. Hyperbolic behaviour of dynamical systems (Paris, 1990).Google Scholar
Hubbard, J. H.. Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz. Topological Methods in Modern Mathematics (Stony Brook, NY, 1991). Publish or Perish, Houston, TX, 1993, pp. 467511.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and Its Applications, 54). Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza.CrossRefGoogle Scholar
Keller, G. and Nowicki, T.. Fibonacci maps re(al)visited. Ergod. Th. & Dynam. Sys. 15(1) (1995), 99120.CrossRefGoogle Scholar
Ledrappier, F.. Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. & Dynam. Sys. 1(1) (1981), 7793.Google Scholar
Lima, Y.. Symbolic dynamics for one dimensional maps with nonuniform expansion. Ann. Inst. H. Poincaré Anal. Non Linéaire 37(3) (2020), 727755.Google Scholar
Luzzatto, S. and Melbourne, I.. Statistical properties and decay of correlations for interval maps with critical points and singularities. Comm. Math. Phys. 320(1) (2013), 2135.CrossRefGoogle Scholar
Lyubich, M. and Milnor, J.. The Fibonacci unimodal map. J. Amer. Math. Soc. 6(2) (1993), 425457.CrossRefGoogle Scholar
Levin, G. and Świątek, G.. Common limits of Fibonacci circle maps. Comm. Math. Phys. 312(3) (2012), 695734.CrossRefGoogle Scholar
Luzzatto, S. and Tucker, W.. Non-uniformly expanding dynamics in maps with singularities and criticalities. Publ. Math. Inst. Hautes Études Sci. 89 (2000), 179226.CrossRefGoogle Scholar
Milnor, J.. Local connectivity of Julia sets: expository lectures. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Cambridge University Press, Cambridge, 2000, pp. 67116.Google Scholar
Milnor, J. and Thurston, W.. On iterated maps of the interval. Dynamical Systems (College Park, MD, 1986–1987) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 465563.Google Scholar
Pedreira, F.. On the behaviour of the singular values of expanding Lorenz maps. PhD Thesis, Universidade Federal da Bahia, 2020.Google Scholar
Przytycki, F.. Lyapunov characteristic exponents are nonnegative. Proc. Amer. Math. Soc. 119(1) (1993), 309317.Google Scholar
Rivera-Letelier, J.. Asymptotic expansion of smooth interval maps. Astérisque 416 (2020), 3363, Quelques aspects de la théorie des systèmes dynamiques: un hommage à Jean-Christophe Yoccoz  II.CrossRefGoogle Scholar
Smania, D.. Puzzle geometry and rigidity: the Fibonacci cycle is hyperbolic. J. Amer. Math. Soc. 20(3) (2007), 629673.CrossRefGoogle Scholar
Tsujii, M.. Positive Lyapunov exponents in families of one-dimensional dynamical systems. Invent. Math. 111(1) (1993), 113137.CrossRefGoogle Scholar