Characterizing finite sets of nonwandering points

TL;DR

This paper characterizes finite sets of nonwandering points for generic diffeomorphisms as uniformly bounded and uses this to describe Morse-Smale diffeomorphisms and reprove results on sinks and sources.

Contribution

It provides a new characterization of finite nonwandering sets and offers a $C^1$ generic description of Morse-Smale diffeomorphisms, also reestablishing known finiteness results.

Findings

Finite nonwandering sets are uniformly bounded.

Characterization of Morse-Smale diffeomorphisms in a $C^1$ generic setting.

Reproof of finiteness of sinks and sources for star diffeomorphisms.

Abstract

We characterize finite sets $S$ of nonwandering points for generic diffeomorphisms $f$ as those which are {\em uniformly bounded}, i.e., there is an uniform bound for small perturbations of the derivative of $f$ along the points in $S$ up to suitable iterates. We use this result to give a $C^1$ generic characterization of the Morse-Smale diffeomorphisms related to the weak Palis conjecture \cite{c}. Furthermore, we obtain another proof of the result by Liao and Pliss about the finiteness of sinks and sources for star diffeomorphisms \cite{l}, \cite{Pl}.