Semiconjugacy to a map of a constant slope

Abstract

Let us consider continuous maps f : X → X and g : Y → Y , where X, Y are compact Hausdorff spaces and φ : X → Y is continuous such that the diagram X f → X ↓φ ↓φ Y g → Y commutes, i.e., φ ◦ f = g ◦ φ. When φ is surjective, we say that f is semiconjugated to g via a map φ and in that case the topological entropy htop(·) satisfies htop(f) ≥ htop(g) [1]. Let X = Y = [0, 1]. A continuous map f : [0, 1] → [0, 1] is said to be piecewise monotone if there are k ∈ N and points 0 = c0 < c1 < · · · < ck−1 < ck = 1 such that f is monotone on each [ci, ci+1], i = 0, . . . , k− 1. We shall say that a piecewise monotone map g has a constant slope s if on each of its pieces of monotonicity it is affine with the slope of absolute value s. In one-dimensional dynamical systems the following interesting result has been proved.