Robust and generic properties for piecewise continuous maps on the interval

TL;DR

This paper develops a new metric for piecewise smooth maps on intervals, demonstrating their robustness and generic properties, including invariant measures, within a Baire space framework.

Contribution

It introduces a novel metric on piecewise r maps, proving their robustness and genericity of certain properties despite the space's incompleteness.

Findings

The metric space of piecewise r maps is a Baire space.

Non-degenerate critical points are robust under this metric.

Generic piecewise Lipschitz maps admit invariant Borel probability measures.

Abstract

We construct an appropriate metric on the collection of piecewise $\mathcal C^r$ maps defined on a compact interval. Although this metric space turns out to be not complete, we show that it is indeed a Baire space. As an application, we prove the robustness of non-degenerate critical points for this class of systems and we show the genericity of piecewise Lipschitz maps that admit an invariant Borel probability measure.