American Institute of Mathematical Sciences

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

Flexibility of Lyapunov exponents for expanding circle maps

Alena Erchenko

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.

Keywords: Flexibility, expanding maps, Lyapunov exponents, circle maps, invariant measures.
Mathematics Subject Classification: Primary: 37E10; Secondary: 37A05.
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

show all references

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

Figure 1.  A representative of the SUSD-circle maps of degree $ 2 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055
[2]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[3]

M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202
[4]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[5]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83
[6]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026
[7]

Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003
[8]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[9]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[10]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (236)
  • HTML views (196)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences