The Derivative of the Conjugacy Between Skew Tent Maps

Abstract

We consider in this article the properties of topological conjugacy of the tent-like maps [Formula: see text], which are piecewise linear and whose graph consists of straight line segments, extending from [Formula: see text] to [Formula: see text] to [Formula: see text], where [Formula: see text] is a parameter. For any point [Formula: see text], we reduce the calculation of the derivative of the conjugacy [Formula: see text] of functions [Formula: see text] and [Formula: see text] to the limit of a recurrently defined sequence, which is defined by [Formula: see text]. In the case [Formula: see text], this result is reduced to a study of some properties of the binary expansion of [Formula: see text].

On one generalization of skew tent maps

“…It is known (see [4]) that all skew tent maps are topologically conjugated to the tent. We have proved in [3] the next fact about the conjugacy of skew tent maps.…”

“…Recall, that the set g −∞ (a) = n 1 g −n (a), where g −n (a) = {x ∈ [0, 1] : g n (x) = a} for all n 1, is called the complete pre-image of a (under the action of the map g). We have made in [3] the following construction, being motivated by Theorem 1.…”

“…Notice that the idea of the proof of Theorem 4 is similar to one of the proof of Theorem 2 in [3]. Nevertheless, it is more interesting, that Theorem 2 can not be generalized to carcass maps.…”

Classification of commutative pairs of surjective maps of interval, one of which is unimodal

“…We will introduced some methods to find the conjugacy from the tent map to a given unimodal map g. For any continuous unimodal map g, which is topologically conjugated to the tent map, we have defined in [4] a g-decomposition (β i ) i 0 of any point x ∈ [0, 1]. This g-decomposition coincides with v −1 -coordinates expansion of a point for the skew tent map f v in our [5]. The next fact follows from [6, Th.…”

Infinite sequence on 2-element alphabet as coordinates of points, determines by unimodal maps

“…We have introduced the notations from this section in our [4] in generalization of the original proof of Theorem 1.…”