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

Abstract

Abstract We consider skew tent maps T α, β (x) such that (α,β)∈[0,1]2 is the turning point of TT α, β , that is, T α, β = β α $\begin{array}{} \frac{{\beta}}{{\alpha}} \end{array} $ x for 0≤ x ≤ α and T α, β (x) = β 1 − α $\begin{array}{} \frac{{\beta}}{1- {\alpha}} \end{array} $ (1−x) for α < x ≤ 1. We denote by M = K(α,β) the kneading sequence of TT α, β and by h(α,β) its topological entropy. For a given kneading squence M we consider equi-kneading, (or equi-topological entropy, or isentrope) curves (α,φ M (α)) such that K(α,φ M (α)) = M . To study the behavior of these curves an auxiliary function Θ M (α,β) is introduced. For this function Θ M (α,φ M (α)) = 0, but it may happen that for some kneading sequences Θ M (α,β) = 0 for some β < φ M (α) with (α,β) still in the dynamically interesting quarter of the unit square. Using Θ M we show that the curves (α,φ 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 M = RLLRC the curve (α,φ M (α)) is not orthogonal to the diagonal. On the other hand, for 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.