Equi-topological entropy curves for skew tent maps in the square

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 $0\le x\le \alpha$ and $T_{\alpha ,\beta }(x)=\frac{\beta }{1-\alpha }(1-x)$ for $\alpha <x\le 1$. We denote by $\underline{M}=K(\alpha ,\beta )$ the kneading sequence of $T_{\alpha ,\beta }$ and by $h(\alpha ,\beta )$ its topological entropy. For a given kneading squence $\underline{M}$ we consider equi-kneading, (or equi-topological entropy, or isentrope) curves $(\alpha ,{\phi }_{\underline{M}}(\alpha ))$ such that $K(\alpha ,{\phi }_{\underline{M}}(\alpha ))=\underline{M}$. To study the behavior of these curves an auxiliary function ${\Theta }_{\underline{M}}(\alpha ,\beta )$ is introduced. For this function ${\Theta }_{\underline{M}}(\alpha ,{\phi }_{\underline{M}}(\alpha ))=0$, but it may happen that for some kneading sequences ${\Theta }_{\underline{M}}(\alpha ,\beta )=0$ for some $\beta <{\phi }_{\underline{M}}(\alpha )$ with $(\alpha ,\beta )$ still in the interesting region. Using ${\Theta }_{\underline{M}}$ we show that the curves $(\alpha ,{\phi }_{\underline{M}}(\alpha ))$ hit the diagonal $\{(\beta ,\beta ):0.5<\beta <1\}$ almost perpendicularly if $(\beta ,\beta )$ is close to $(1,1)$. Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for $\underline{M}=RLLRC$ the curve $(\alpha ,{\phi }_{\underline{M}}(\alpha ))$ is not orthogonal to the diagonal. On the other hand, for $\underline{M}=RLC$ it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J.C. Marcuard, M. Misiurewicz and E. Visinescu.