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April  2020, 40(4): 1989-2009. doi: 10.3934/dcds.2020102

Isentropes and Lyapunov exponents

Zoltán Buczolich , and  Gabriella Keszthelyi

Department of Analysis, ELTE Eötvös Loránd University, Pázmány Péter Sétány 1/c, 1117, Budapest, Hungary, ZB's ORCID ID: 0000-0001-5481-8797

* Corresponding author

Received  April 2018 Revised  July 2019 Published  January 2020

Fund Project: ZB was supported by the Hungarian National Foundation for Scientific Research Grant 124003. During the preparation of this paper this author was a visiting researcher at the Rényi Institute.
GK was supported by the Hungarian National Foundation for Scientific Research Grant 124749.

We consider skew tent maps $ T_{ { \alpha }, { \beta }}(x) $ such that $ ( \alpha, \beta)\in[0, 1]^{2} $ is the turning point of $ T _{\alpha, \beta} $, that is, $ T_{ { \alpha }, { \beta }} = \frac{ { \beta }}{ { \alpha }}x $ for $ 0\leq x \leq { \alpha } $ and $ T_{ { \alpha }, { \beta }}(x) = \frac{ { \beta }}{1- { \alpha }}(1-x) $ for $ { \alpha }<x\leq 1 $. We denote by $ \underline{M} = K( \alpha, \beta) $ the kneading sequence of $ T _{\alpha, \beta} $, by $ h( \alpha, \beta) $ its topological entropy and $ \Lambda = \Lambda_{\alpha, \beta} $ denotes its Lyapunov exponent. For a given kneading squence $ \underline{M} $ we consider isentropes (or equi-topological entropy, or equi-kneading curves), $ ( \alpha, \Psi_{ \underline{M}}( \alpha)) $ such that $ K( \alpha, \Psi_{ \underline{M}}( \alpha)) = \underline{M} $. On these curves the topological entropy $ h( \alpha, \Psi_{ \underline{M}}( \alpha)) $ is constant.

We show that $ \Psi_{ \underline{M}}'( \alpha) $ exists and the Lyapunov exponent $ \Lambda_{\alpha, \beta} $ can be expressed by using the slope of the tangent to the isentrope. Since this latter can be computed by considering partial derivatives of an auxiliary function $ \Theta_{ \underline{M}} $, a series depending on the kneading sequence which converges at an exponential rate, this provides an efficient new method of finding the value of the Lyapunov exponent of these maps.

Keywords: Skew tent map, topological entropy, isentrope, invariant measure, Lyapunov exponent, Markov partition.
Mathematics Subject Classification: Primary: 37B25; Secondary: 28D20, 37B40, 37E05.
Citation: Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102
References:
[1]

V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711.  doi: 10.1088/0951-7715/21/4/003.  Google Scholar

[2]

V. Baladi and D. Smania, Smooth deformations of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst., 23 (2009), 685-703.  doi: 10.3934/dcds.2009.23.685.  Google Scholar

[3]

L. Billings and E. M. Bollt, Probability density functions of some skew tent maps, Chaos Solitons Fractals, 12 (2001), 365-376.  doi: 10.1016/S0960-0779(99)00204-0.  Google Scholar

[4]

A. Boyarsky and P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-1-4612-2024-4.  Google Scholar

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H. Bruin and S. van Strien, On the structure of isentropes of polynomial maps, Dyn. Syst., 28 (2013), 381-392.  doi: 10.1080/14689367.2013.822458.  Google Scholar

[6]

Z. Buczolich and G. Keszthelyi, Equi-topological entropy curves for skew tent maps in the square, Math. Slovaca, 67 (2017), 1577-1594.  doi: 10.1515/ms-2017-0072.  Google Scholar

[7]

Z. Buczolich and G. Keszthelyi, Tangents of Isentropes of skew tent maps in the square, (in preparation). Google Scholar

[8]

P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics, 1. Birkhäuser, Boston, Mass., 1980.  Google Scholar

[9]

G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385.  Google Scholar

[10]

D. J. Lai and G. R. Chen, On statistical properties of the Lyapunov exponent of the generalized skew tent map, Stochastic Anal. Appl., 20 (2002), 375-388.  doi: 10.1081/SAP-120003440.  Google Scholar

[11]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1974), 481-488.  doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar

[12]

M. C. Mackey and M. Tyran-Kamińska, Central limit theorem behavior in the skew tent map, Chaos Solitons Fractals, 38 (2008), 789-805.  doi: 10.1016/j.chaos.2007.01.013.  Google Scholar

[13]

J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems (College Park, MD, 1986–87), Lecture Notes in Math., Springer, Berlin, 1342 (1988), 465-563.  doi: 10.1007/BFb0082847.  Google Scholar

[14]

J. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, With an appendix by Adrien Douady and Pierrette Sentenac. Comm. Math. Phys., 209 (2000), 123-178.  doi: 10.1007/s002200050018.  Google Scholar

[15]

M. Misiurewicz and E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. Henri Poincaré, Probab. Stat., 27 (1991), 125-140.   Google Scholar

[16]

A. Radulescu, The connected isentropes conjecture in a space of quartic polynomials, Discrete Contin. Dyn. Syst., 19 (2007), 139-175.  doi: 10.3934/dcds.2007.19.139.  Google Scholar

show all references

References:
[1]

V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711.  doi: 10.1088/0951-7715/21/4/003.  Google Scholar

[2]

V. Baladi and D. Smania, Smooth deformations of piecewise expanding unimodal maps, Discrete Contin. Dyn. Syst., 23 (2009), 685-703.  doi: 10.3934/dcds.2009.23.685.  Google Scholar

[3]

L. Billings and E. M. Bollt, Probability density functions of some skew tent maps, Chaos Solitons Fractals, 12 (2001), 365-376.  doi: 10.1016/S0960-0779(99)00204-0.  Google Scholar

[4]

A. Boyarsky and P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension, Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-1-4612-2024-4.  Google Scholar

[5]

H. Bruin and S. van Strien, On the structure of isentropes of polynomial maps, Dyn. Syst., 28 (2013), 381-392.  doi: 10.1080/14689367.2013.822458.  Google Scholar

[6]

Z. Buczolich and G. Keszthelyi, Equi-topological entropy curves for skew tent maps in the square, Math. Slovaca, 67 (2017), 1577-1594.  doi: 10.1515/ms-2017-0072.  Google Scholar

[7]

Z. Buczolich and G. Keszthelyi, Tangents of Isentropes of skew tent maps in the square, (in preparation). Google Scholar

[8]

P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics, 1. Birkhäuser, Boston, Mass., 1980.  Google Scholar

[9]

G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385.  Google Scholar

[10]

D. J. Lai and G. R. Chen, On statistical properties of the Lyapunov exponent of the generalized skew tent map, Stochastic Anal. Appl., 20 (2002), 375-388.  doi: 10.1081/SAP-120003440.  Google Scholar

[11]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1974), 481-488.  doi: 10.1090/S0002-9947-1973-0335758-1.  Google Scholar

[12]

M. C. Mackey and M. Tyran-Kamińska, Central limit theorem behavior in the skew tent map, Chaos Solitons Fractals, 38 (2008), 789-805.  doi: 10.1016/j.chaos.2007.01.013.  Google Scholar

[13]

J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems (College Park, MD, 1986–87), Lecture Notes in Math., Springer, Berlin, 1342 (1988), 465-563.  doi: 10.1007/BFb0082847.  Google Scholar

[14]

J. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, With an appendix by Adrien Douady and Pierrette Sentenac. Comm. Math. Phys., 209 (2000), 123-178.  doi: 10.1007/s002200050018.  Google Scholar

[15]

M. Misiurewicz and E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. Henri Poincaré, Probab. Stat., 27 (1991), 125-140.   Google Scholar

[16]

A. Radulescu, The connected isentropes conjecture in a space of quartic polynomials, Discrete Contin. Dyn. Syst., 19 (2007), 139-175.  doi: 10.3934/dcds.2007.19.139.  Google Scholar

Figure 1.  Tangents to isentropes computed from $ \gamma $ and from $ \Theta $
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Figure 2.  More tangents to isentropes computed from $ \gamma $ and from $ \Theta $
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Figure 3.  Isentropes and tangents computed from $ \gamma $
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Figure 4.  Illustration for the proofs of Proposition 12 and Theorem 2
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Table 1.  Tangents calculated from $ \Theta $ and $ \gamma $
$ \alpha $ $ \beta $ $ \gamma $ $ \Psi_{\underline{M}}' $–$ \gamma $ $ \Psi_{\underline{M}}' $–$ \Theta $
.3 .8 .20444 -.36406 -.36452
.49 .56 .30996 -.40344 -.4244
.5 .7 .27034 -.64303 -.64064
.5 .8 .26918 -.73861 -.73739
.6 .75 .35597 -.76258 -.76132
.6 .9 .47736 -.4599 -.45991
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$ \alpha $ $ \beta $ $ \gamma $ $ \Psi_{\underline{M}}' $–$ \gamma $ $ \Psi_{\underline{M}}' $–$ \Theta $
.3 .8 .20444 -.36406 -.36452
.49 .56 .30996 -.40344 -.4244
.5 .7 .27034 -.64303 -.64064
.5 .8 .26918 -.73861 -.73739
.6 .75 .35597 -.76258 -.76132
.6 .9 .47736 -.4599 -.45991
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