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Lyapunov spectrum for Hénon-like maps at the first bifurcation
Published online by Cambridge University Press: 10 November 2016
Abstract
For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.
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References
Barreira, L. and Iommi, G.. Multifractal analysis and phase transitions for hyperbolic and parabolic horseshoes. Israel J. Math.
181 (2011), 347–479.Google Scholar
Barreira, L. and Schmeling, J.. Sets of ‘non-typical’ points have full Hausdorff dimension and full topological entropy. Israel J. Math.
116 (2000), 29–70.Google Scholar
Bedford, E. and Smillie, J.. Real polynomial diffeomorphisms with maximal entropy: II. Small Jacobian. Ergod. Th. & Dynam. Sys.
26 (2006), 1259–1283.Google Scholar
Benedicks, M. and Carleson, L.. The dynamics of the Hénon map. Ann. of Math. (2)
133 (1991), 73–169.Google Scholar
Benedicks, M. and Viana, M.. Solution of the basin problem for Hénon-like attractors. Invent. Math.
143 (2001), 375–434.Google Scholar
Cao, Y., Luzzatto, S. and Rios, I.. The boundary of hyperbolicity for Hénon-like families. Ergod. Th. & Dynam. Sys.
28 (2008), 1049–1080.Google Scholar
Chung, Y. M.. Birkhoff spectra for one-dimensional maps with some hyperbolicity. Stoch. Dyn.
10 (2010), 53–75.Google Scholar
Chung, Y. M. and Takahasi, H.. Multifractal formalism for Benedicks–Carleson quadratic maps. Ergod. Th. & Dynam. Sys.
34 (2014), 1116–1141.Google Scholar
Devaney, R. and Nitecki, Z.. Shift automorphisms in the Hénon mapping. Comm. Math. Phys.
67 (1979), 137–146.Google Scholar
Eizenberg, A., Kifer, Y. and Weiss, B.. Large deviations for ℤ
d
-actions. Comm. Math. Phys.
164 (1994), 433–454.Google Scholar
Gelfert, K., Przytycki, F. and Rams, M.. On the Lyapunov spectrum for rational maps. Math. Ann.
348 (2010), 965–1004.Google Scholar
Gelfert, K. and Rams, M.. The Lyapunov spectrum of some parabolic systems. Ergod. Th. & Dynam. Sys.
29 (2009), 919–940.Google Scholar
Iommi, G. and Todd, M.. Dimension theory for multimodal maps. Ann. Henri Poincaré
12 (2011), 591–620.Google Scholar
Johansson, A., Jordan, T., Öberg, A. and Pollicott, M.. Multifractal analysis of non-uniformly hyperbolic systems. Israel J. Math.
177 (2010), 125–144.Google Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci.
51 (1980), 137–173.Google Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.Google Scholar
Nakaishi, K.. Multifractal formalism for some parabolic maps. Ergod. Th. & Dynam. Sys.
20 (2000), 843–857.Google Scholar
Olsen, L.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl.
82 (2003), 1591–1649.Google Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago, 1997.Google Scholar
Pesin, Y. and Weiss, H.. A multifracatal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys.
86 (1997), 233–275.Google Scholar
Przytycki, F. and Rivera-Letelier, J.. Nice inducing schemes and the thermodynamics of rational maps. Comm. Math. Phys.
70 (2011), 661–707.Google Scholar
Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Mat.
9 (1978), 83–87.Google Scholar
Senti, S. and Takahasi, H.. Equilibrium measures for the Hénon map at the first bifurcation. Nonlinearity
26 (2013), 1719–1741.Google Scholar
Senti, S. and Takahasi, H.. Equilibrium measures for the Hénon map at the first bifurcation: uniqueness and geometric/statistical properties. Ergod. Th. & Dynam. Sys.
36 (2016), 215–255.Google Scholar
Sigmund, K.. On dynamical systems with the specification property. Trans. Amer. Math. Soc.
190 (1974), 285–299.Google Scholar
Takahasi, H.. Abundance of nonuniform hyperbolicity in bifurcations of surface endomorphisms. Tokyo J. Math.
34 (2011), 53–113.CrossRefGoogle Scholar
Takahasi, H.. Prevalent dynamics at the first bifurcation of Hénon-like families. Comm. Math. Phys.
312 (2012), 37–85.Google Scholar
Takahasi, H.. Equilibrium measures at temperature zero for Hénon-like maps at the first bifurcation. SIAM J. Appl. Dyn. Syst.
15 (2016), 106–124.Google Scholar
Takahasi, H.. Prevalence of non-uniform hyperbolicity at the first bifurcation of Hénon-like families. Preprint. Available at http://arxiv.org/abs/1308.4199.Google Scholar
Takens, F. and Verbitskiy, E.. On the variational principle for the topological entropy of certain non-compact sets. Ergod. Th. & Dynam. Sys.
23 (2003), 317–348.Google Scholar
Urbański, M. and Wolf, C.. Ergodic theory of parabolic horseshoes. Comm. Math. Phys.
281 (2008), 711–751.Google Scholar
Walters, P.. An Introduction to Ergodic Theory
(Graduate Texts in Mathematics, 79)
. Springer, New York, 1982.Google Scholar
Wang, Q. D. and Young, L.-S.. Strange attractors with one direction of instability. Comm. Math. Phys.
218 (2001), 1–97.Google Scholar
Weiss, H.. The Lyapunov spectrum for conformal expanding maps and Axiom A surface diffeomorphisms. J. Stat. Phys.
95 (1999), 615–632.Google Scholar
Young, L.-S.. Dimension, entropy and Lyapunov exponents. Ergod. Th. & Dynam. Sys.
2 (1982), 109–124.Google Scholar