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The Lyapunov spectrum of some parabolic systems



We study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.



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[1]Barreira, L. and Saussol, B.. Variational principles and mixed multifractal spectra. Trans. Amer. Math. Soc. 353 (2001), 39193944.
[2]Barreira, L. and Schmeling, J.. Sets of ‘non-typical’ points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116 (2000), 2970.
[3]Denker, M. and Urbański, M.. On the existence of conformal measures. Trans. Amer. Math. Soc. 328 (1991), 563578.
[4]Gelfert, K. and Rams, M.. Geometry of limit sets for expansive Markov systems, Trans. Amer. Math. Soc. to appear.
[5]Hofbauer, F. and Raith, P.. The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval. Canad. Math. Bull. 35 (1992), 8498.
[6]Jenkinson, O.. Rotation, entropy, and equilibrium states. Trans. Amer. Math. Soc. 353 (2001), 37133739.
[7]Kesseböhmer, M. and Stratmann, B. O.. Stern–Brocot pressure and multifractal spectra in ergodic theory of numbers. Stoch. Dyn. 4 (2004), 7784.
[8]de Melo, W. and van Strien, S.. One-dimensional Dynamics. Springer, Berlin, 1993.
[9]Nakaishi, K.. Multifractal formalism for some parabolic maps. Ergod. Th. & Dynam. Sys. 20 (2000), 843857.
[10]Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications (Chicago Lectures in Mathematics). The University of Chicago Press, Chicago, 1998.
[11]Pfister, C.-E. and Sullivan, W.. On the topological entropy of saturated sets. Ergod. Th. & Dynam. Sys. 27 (2007), 929956.
[12]Pomeau, Y. and Manneville, P.. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980), 189197.
[13]Takens, F. and Verbitskiy, E.. On the variational principle for the topological entropy of certain non-compact sets. Ergod. Th. & Dynam. Sys. 23 (2003), 317348.
[14]Urbański, M.. Parabolic Cantor sets. Fund. Math. 151 (1996), 241277.
[15]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121153.
[16]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, Berlin, 1981.
[17]Wijsman, R.. Convergence of sequence of convex sets, cones, and functions. II. Trans. Amer. Math. Soc. 123 (1966), 3245.
[18]Yuri, M.. Thermodynamic formalism for certain nonhyperbolic maps. Ergod. Th. & Dynam. Sys. 19 (1999), 13651378.


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