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

TL;DR

This paper investigates the geometric properties of equi-topological entropy curves for skew tent maps, revealing their near-perpendicular intersection with the diagonal and analyzing conditions for orthogonality.

Contribution

It introduces an auxiliary function to study the shape of equi-kneading curves and demonstrates their intersection behavior with the diagonal, addressing a question posed by M. Misiurewicz.

Findings

Curves hit the diagonal almost perpendicularly near (1,1)

Not all equi-kneading curves are exactly orthogonal to the diagonal

Orthogonality depends on the specific kneading sequence

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 $T_ {{\alpha}, {\beta}}$ 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 $({\alpha}, \varphi_{{\underline{M}}}({\alpha}))$ such that $K({\alpha}, {\varphi}_{{\underline{M}}}({\alpha}))= {\underline{M}}$. To study the behavior of these curves an auxiliary function $ {\Theta}_{{\underline{M}}}({\alpha}, {\beta})$ is introduced. For this function $ {\Theta}_{{\underline{M}}}({\alpha}, \varphi_{{\underline{M}}}({\alpha}))=0$, but it may happen that for some kneading sequences $\Theta_{{\underline{M}}}({\alpha}, {\beta})=0$ for some $ {\beta}< \varphi_{{\underline{M}}}({\alpha})$ with $({\alpha}, {\beta})$ still in the interesting region. Using $ {\Theta}_{{\underline{M}}}$ we show that the curves $({\alpha},\varphi_{{\underline{M}}}({\alpha}))$ 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 $(\alpha, {\varphi}_{{\underline{M}}}({\alpha}))$ 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.