Isentropes and Lyapunov exponents

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.