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

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 $T_ {α, β}$ and by $h(α, β)$ its topological entropy. For a given kneading squence $ {\underline{M}}$ we consider equi-kneading, (or equi-topological entropy, or isentrope) curves $(α, φ_{\underline{M}}(α))$ such that $K(α, φ_{\underline{M}}(α))= {\underline{M}}$. To study the behavior of these curves an auxiliary function $ Θ_{\underline{M}}(α, β)$ is introduced. For this function $ Θ_{\underline{M}}(α, φ_{\underline{M}}(α))=0$, but it may happen that for some kneading sequences $Θ_{\underline{M}}(α, β)=0$ for some $ β< φ_{\underline{M}}(α)$ with $(α, β)$ still in the interesting region. Using $ Θ_{\underline{M}}$ we show that the curves $(α,φ_{\underline{M}}(α))$ hit the diagonal $\{(β, β): 0.5< β<1 \}$ almost perpendicularly if $(β, β)$ 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 $(α, φ_{\underline{M}}(α))$ 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.