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

show all references

##### 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
 [1] Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3797-3816. doi: 10.3934/dcds.2021017 [2] Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159 [3] Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249 [4] Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295 [5] Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555 [6] Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363 [7] Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012 [8] Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . doi: 10.3934/dcds.2011.31.545 [9] Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 [10] Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192 [11] Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201 [12] Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827 [13] Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021131 [14] Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048 [15] Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197 [16] Sanjeeva Balasuriya. Uncertainty in finite-time Lyapunov exponent computations. Journal of Computational Dynamics, 2020, 7 (2) : 313-337. doi: 10.3934/jcd.2020013 [17] Donald Ornstein, Benjamin Weiss. Entropy is the only finitely observable invariant. Journal of Modern Dynamics, 2007, 1 (1) : 93-105. doi: 10.3934/jmd.2007.1.93 [18] Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433 [19] James P. Kelly, Kevin McGoff. Entropy conjugacy for Markov multi-maps of the interval. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2071-2094. doi: 10.3934/dcds.2020353 [20] Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1

2020 Impact Factor: 1.392