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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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