Isentropes and Lyapunov exponents

TL;DR

This paper introduces a new method to efficiently compute Lyapunov exponents for skew tent maps by analyzing isentropes and their tangent slopes, leveraging kneading sequences and an auxiliary function.

Contribution

It develops a novel approach linking the tangent slope of isentropes to Lyapunov exponents using an auxiliary function with exponential convergence.

Findings

Derived the existence of the derivative of isentropes.

Expressed Lyapunov exponents via tangent slopes of isentropes.

Presented an efficient computational method using a convergent series.

Abstract

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.