Definition and applications of the ascent-probability distribution in one-dimensional maps

This paper introduces the discrete distribution of ascent probabilities {P_n}_n=0, generalizing the concept of rotation number, already being defined in one-dimensional unimodal maps. The map domain is partitioned into subintervals {I_n}_n=0, each one containing orbits with their route characterized by a specific number of n successive ascents before a descent occurs. Then, P_n is defined as the probability to have an orbit performing n successive ascents, and equals the portion of the invariant measure within I_n. The rotation number is found to be equal to = 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 {P_n}_n=0 within the chaotic zone. A particular set of nonlinear parameter values {p(n)}_n=3 is affiliated to the concept of ascent probabilities: For each probability P_n, n = 0, 1, 2, there is a lower limit of the nonlinear parameter values, p_(n+1), so that, P_n = 0 ∀ p ≤ p_(n+1). The set {p(n)} _n=3 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, {u_n} _n=0, of the invariant density S, and of the set of the ascents probabilities, {P_n}_n=0, 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.