# American Institute of Mathematical Sciences

May  2019, 39(5): 2325-2342. doi: 10.3934/dcds.2019098

## Flexibility of Lyapunov exponents for expanding circle maps

 Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received  May 2017 Revised  July 2018 Published  January 2019

Let $g$ be a smooth expanding map of degree $D$ which maps a circle to itself, where $D$ is a natural number greater than $1$. It is known that the Lyapunov exponent of $g$ with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to $\log D$ which, in addition, is less than or equal to the Lyapunov exponent of $g$ with respect to the measure of maximal entropy. Moreover, the equalities can only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree $D$ on a circle.

Citation: Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
##### References:
 [1] J. Bochi, A. Katok and F. Rodrigues Hertz, Flexibility of Lyapunov exponents among conservative diffeomorphisms, preprint. Google Scholar [2] A. Boyarsky and M. Scarowsky, On a class of transformations which have unique absolutely continuous invariant measures, Trans. Amer. Math. Soc., 255 (1979), 243-262.  doi: 10.1090/S0002-9947-1979-0542879-2.  Google Scholar [3] A. Erchenko and A. Katok, Flexibility of entropies for surfaces of negative curvature, to appear in Israel J. Math., arXiv:1710.00079. Google Scholar [4] A. Góra and A. Boyarsky, Compactness of invariant densities for families of expanding, piecewise monotonic transformations, Canad. J. Math., 41 (1989), 855-869.  doi: 10.4153/CJM-1989-039-8.  Google Scholar [5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar [6] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488.  doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar [7] M. Qian, J.-S. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01954-8.  Google Scholar [8] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.  doi: 10.1007/BF02584795.  Google Scholar [9] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153.  doi: 10.1090/S0002-9947-1978-0466493-1.  Google Scholar

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##### References:
 [1] J. Bochi, A. Katok and F. Rodrigues Hertz, Flexibility of Lyapunov exponents among conservative diffeomorphisms, preprint. Google Scholar [2] A. Boyarsky and M. Scarowsky, On a class of transformations which have unique absolutely continuous invariant measures, Trans. Amer. Math. Soc., 255 (1979), 243-262.  doi: 10.1090/S0002-9947-1979-0542879-2.  Google Scholar [3] A. Erchenko and A. Katok, Flexibility of entropies for surfaces of negative curvature, to appear in Israel J. Math., arXiv:1710.00079. Google Scholar [4] A. Góra and A. Boyarsky, Compactness of invariant densities for families of expanding, piecewise monotonic transformations, Canad. J. Math., 41 (1989), 855-869.  doi: 10.4153/CJM-1989-039-8.  Google Scholar [5] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.  Google Scholar [6] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488.  doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar [7] M. Qian, J.-S. Xie and S. Zhu, Smooth Ergodic Theory for Endomorphisms, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-01954-8.  Google Scholar [8] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat., 9 (1978), 83-87.  doi: 10.1007/BF02584795.  Google Scholar [9] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153.  doi: 10.1090/S0002-9947-1978-0466493-1.  Google Scholar
A representative of the SUSD-circle maps of degree $2$
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