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

We consider skew tent maps $T_{\alpha ,\beta }(x)$ such that $({\alpha}, {\beta})\in[0,1]^{2}$istheturningpointof$T {_ {{\alpha}, {\beta}}}$, 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$\mathrm{.}Wedenoteby$ {\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 $\underset{‾}{M}$ we consider equi-kneading, (or equi-topological entropy, or isentrope) curves $(\alpha ,{\phi }_{\underset{‾}{M}}(\alpha ))$ 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})$isintroduced\mathrm{.}Forthisfunction$ {\Theta}{{\underline{M}}}({\alpha}, \varphi{{\underline{M}}}({\alpha}))=0$, but it may happen that for some kneading sequences ${\mathrm{\Theta }}_{\underset{‾}{M}}(\alpha ,\beta )=0$ for some $ {\beta}< \varphi_{{\underline{M}}}({\alpha})$with$({\alpha}, {\beta})$ still in the interesting region. Using ${\mathrm{\Theta }}_{\underset{‾}{M}}$ we show that the curves $(\alpha ,{\phi }_{\underset{‾}{M}}(\alpha ))$ hit the diagonal ${({\beta}, {\beta}): 0.5< {\beta}<1 }$almostperpendicularlyif$({\beta}, {\beta})$ 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 $\underset{‾}{M}=RLLRC$ the curve $(\alpha, {\varphi}_{{\underline{M}}}({\alpha}))$ is not orthogonal to the diagonal. On the other hand, for $\underset{‾}{M}=RLC$ 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.