DEFINITION AND APPLICATIONS OF THE ASCENT-PROBABILITY DISTRIBUTION IN ONE-DIMENSIONAL MAPS

Abstract

This paper introduces the discrete distribution of ascent probabilities [Formula: see text], generalizing the concept of rotation number ω, already being defined in one-dimensional unimodal maps. The map domain is partitioned into subintervals [Formula: see text], each one containing orbits with their route characterized by a specific number of n successive ascents before a descent occurs. Then, Pn is defined as the probability to have an orbit performing n successive ascents, and equals the portion of the invariant measure within In. The rotation number is found to be equal to ω = P0 = P1, that is the portion of invariant measure within I0 or I1. Some significant applications of this relation concern (i) the rotation number dependence on the nonlinear parameter p, (ii) the analytical derivation of the rotation number, given the invariant density explicit expression, (iii) an easy computation of the rotation number that characterizes a periodic window. Moreover, the dependence of the ascent probabilities on the nonlinear parameter p is examined. Emphasis is placed on the discrete distribution of the ascent probabilities [Formula: see text] within the chaotic zone. A particular set of nonlinear parameter values [Formula: see text] is affiliated to the concept of ascent probabilities: For each probability Pn, n = 0, 1, 2, …, there is a lower limit of the nonlinear parameter values, p(n+1), so that, Pn = 0 ∀ p ≤ p(n+1). The set [Formula: see text] is analytically determined, and its specific arrangement in the chaotic zone is studied. Finally, the "u-S-P equivalence" between the triplet of the set of the fixed point and its preimages, [Formula: see text], of the invariant density S, and of the set of the ascents probabilities, [Formula: see text], is formulated. In particular, we show that each component of this triplet can be estimated whenever the other two components are given. Applications in the case of Logistic map are thoroughly examined.