On generic $G$-prevalent properties of $C^{r}$ diffeomorphisms of $\mathbf{S}^{1}$ and a quantitative K-S theorem

TL;DR

This paper introduces a generalized notion of prevalence for group actions on convex sets, establishing a quantitative Kupka-Smale theorem and results on rotation numbers for $C^r$ diffeomorphisms of the circle, extending classical dynamical systems theory.

Contribution

It develops a new framework for $G$-prevalence replacing translation structure, and proves a quantitative Kupka-Smale theorem along with rotation number results in this setting.

Findings

Established a $G$-prevalence concept for group actions.

Proved a quantitative Kupka-Smale theorem for $C^r$ circle diffeomorphisms.

Derived results on rotation numbers extending Yoccoz's work.

Abstract

We will consider a convex unbounded set and certain group of actions $G$ on this set. This will substitute the translation (by adding) structure usually consider in the classical setting of prevalence. In this way we will be able to define the meaning of $G$-prevalent set. In this setting we will show a kind of quantitative Kupka-Smale Theorem and also a result about rotation numbers which was first consider by J.-C. Yoccoz (and, also by M. Tsujii).