Dynamical behavior of a parametrized family of one-dimensional maps

Abstract

We investigate the dynamics of the maps fµ,w (x) := xµ sin(w ln(x)) with µ > 1 (and odd continuation). The first chapter describes how a family of one-dimensional maps fµ,w appears in the context of return maps associated to homoclinic orbits for ODEs. Corresponding to the shape of graph of fµ,w, we introduce so-called "flat" intervals containing exactly one maximum or minimum. We shall also use the expression "steep" for intervals containing exactly one zero point of fµ,w. Then we construct an open set of points with orbits staying entirely in the "flat" intervals in chapter three. In the fourth chapter, it is proved that there exist some points whose orbits stay totally within the "steep" intervals. Then, to orbits (fj (x)) of fµ,w we associate a symbol sequence (sj) = (signfj (x)) = (+1, 1, 1, +1, ...), and we show that the measure of the set of points which follow such symbol sequences is zero. In the last chapter, it is shown that there exist some points whose orbits travel regularly from "flat" intervals to "steep" intervals, then from "steep" to "flat" intervals and so on. To such orbits of fµ,w we associate a symbol sequence (L, R, R, L, ...) , indicating whether the iterates of points are to the left or to the right of corresponding maxima of fµ,w, and finally the Lebesgue measure of the set of these points is shown to be zero.