Isentropes and Lyapunov exponents

Abstract

We consider skew tent maps $T_{α, β}(x)$ such that $( α, β)\in[0,1]^{2}$ is the turning point of $T {_ { α, β}}$, that is, $T_{α, β}=\frac{β}{α}x$ for $0\leq x \leq α$ and $T_{α, β}(x)=\frac{β}{1- α}(1-x)$ for $ α<x\leq 1$. We denote by $ {\underline {M}}=K( α, β)$ the kneading sequence of $T {_ { α, β}}$, by $h( α, β)$ its topological entropy and $Λ=Λ_{α,β}$ denotes its Lyapunov exponent. For a given kneading squence $ {\underline {M}}$ we consider isentropes (or equi-topological entropy, or equi-kneading curves), $( α,Ψ_{\underline {M}}( α))$ such that $K( α,Ψ_{\underline {M}}( α))= {\underline {M}}$. On these curves the topological entropy $h( α,Ψ_{\underline {M}}( α))$ is constant. We show that $Ψ_{\underline {M}}'( α)$ exists and the Lyapunov exponent $Λ_{α,β}$ 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 $ { Θ}_{\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.