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Isentropes and Lyapunov exponents

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    * Corresponding author 

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

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  • 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.

    Mathematics Subject Classification: Primary: 37B25; Secondary: 28D20, 37B40, 37E05.

    Citation:

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  • Figure 1.  Tangents to isentropes computed from $ \gamma $ and from $ \Theta $

    Figure 2.  More tangents to isentropes computed from $ \gamma $ and from $ \Theta $

    Figure 3.  Isentropes and tangents computed from $ \gamma $

    Figure 4.  Illustration for the proofs of Proposition 12 and Theorem 2

    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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  • [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [7] Z. Buczolich and G. Keszthelyi, Tangents of Isentropes of skew tent maps in the square, (in preparation).
    [8] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Progress in Physics, 1. Birkhäuser, Boston, Mass., 1980.
    [9] G. Keller, Stochastic stability in some chaotic dynamical systems, Monatsh. Math., 94 (1982), 313-333.  doi: 10.1007/BF01667385.
    [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.
    [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.
    [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.
    [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.
    [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.
    [15] M. Misiurewicz and E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. Henri Poincaré, Probab. Stat., 27 (1991), 125-140. 
    [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.
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