Statistical properties of invariant graphs in piecewise affine discontinuous forced systems

Abstract

When a contractive map is forced by a chaotic discontinuous system, the asymptotic response function that defines the attracting invariant set can be highly irregular. In this context, it is natural to ask whether the invariant distributions of the base and factor systems share the same characteristics and in particular, whether the factor distribution of an absolutely continuous measure in the base can be absolutely continuous. Here, we address this question in a basic example of linear real contractions forced by (generalized) baker's maps and we prove absolute continuity for almost every value of the factor contraction rate.

Attracting graphs of skew products with non-contracting fiber maps

“…Thus Theorem 3 holds in this setting. As far as we know, the study of ergodic properties for these skew products is still quite incipient and there are very few available results, see [19] and also [6] for some specific cases. We refer to [10] for a bifurcation setting.…”

The variation of invariant graphs in forced systems

“…On the other hand, the total variation of φ becomes infinite when γ is sufficiently large [10]. A sync function of bounded variation makes it easier to show that the factor statistics inherits absolute continuity of the forcing statistics [11]. In any case, topological feature of f −1 can also affect regularity properties of φ.…”