Minimal set of periods for continuous self-maps of the eight space

Jaume Llibre,Ana Sá

doi:10.1186/s13663-020-00687-9

Abstract

Let G_{k} be a bouquet of circles, i.e., the quotient space of the interval [0,k] obtained by identifying all points of integer coordinates to a single point, called the branching point of G_{k}. Thus, G_{1} is the circle, G_{2} is the eight space, and G_{3} is the trefoil. Let f: G_{k} to G_{k} be a continuous map such that, for k>1, the branching point is fixed.If operatorname{Per}(f) denotes the set of periods of f, the minimal set of periods of f, denoted by operatorname{MPer}(f), is defined as bigcap_{gsimeq f} operatorname{Per}(g) where g:G_{k}to G_{k} is homological to f.The sets operatorname{MPer}(f) are well known for circle maps. Here, we classify all the sets operatorname{MPer}(f) for self-maps of the eight space.

Highlights

Introduction and statement of the resultsIn dynamical systems it is often the case that topological information can be used to study qualitative or quantitative properties of the system
This work deals with the problem of determining the set of periods of the periodic orbits of a map given the homology class of the map
A finite graph G is a topological space formed by a finite set of points V and a finite set of open arcs in such a way that each open arc is attached by its endpoints to vertices

Summary

Introduction

Introduction and statement of the resultsIn dynamical systems it is often the case that topological information can be used to study qualitative or quantitative properties of the system. Our main objective is to characterize the minimal sets of periods MPer(f ) for graph maps f : Gi → Gi with the branching point a fixed point for i = 2, 3.

Results

Conclusion