Topological Bifurcation Structure Of One-Parameter Families Of $C^1$ Unimodal Maps

TL;DR

This paper analyzes the bifurcation structure of one-parameter $C^1$ unimodal maps, showing that topologically it resembles quadratic polynomial families despite potential complexity, and introduces methods to simplify and understand these structures.

Contribution

It demonstrates that irreducible components of bifurcation diagrams can be defined under mild differentiability conditions, extending classical results to less smooth maps.

Findings

Bifurcation structures are topologically similar to quadratic families.

Irreducible components can be characterized by symbolic conditions.

Separation theorem allows isolating connected components in complex bifurcation diagrams.

Abstract

We consider the bifurcation structure of one-parameter families of unimodal maps whose differentiability is only $C^1$. The structure of its bifurcation diagram can be a very wild one in such case. However we prove that in a certain topological sense, the structure is the same as that of the standard family of quadratic polynomials. In the case of families of polynomials, irreducible component of the bifurcation diagram can be defined naturally by dividing by the polynomials corresponding to lower periods. We show that such a irreducible component can be defined even if the maps and the family satisfy only a very mild differentiability condition. By removing components of lower periods, the structure of the bifurcation diagram becomes a considerably simplified one. We prove that the symbolic condition for the irreducible components is exactly the same as that of the standard family of quadratic polynomials. When we consider families of maps without strict conditions, the bifurcation diagrams may have infinitely many wild components. We show that such a situation does not affect the irreducible component essentially by proving a separation theorem for compact set in the plane which asserts that a given connected component can be cut out from the compact set by a curve.