American Institute of Mathematical Sciences

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

Isentropes and Lyapunov exponents

 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.

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:

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References:
Tangents to isentropes computed from $\gamma$ and from $\Theta$
More tangents to isentropes computed from $\gamma$ and from $\Theta$
Isentropes and tangents computed from $\gamma$
Illustration for the proofs of Proposition 12 and Theorem 2
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
 $\alpha$ $\beta$ $\gamma$ $\Psi_{\underline{M}}'$–$\gamma$ $\Psi_{\underline{M}}'$–$\Theta$ .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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