# The bifurcation set as a topological invariant for one-dimensional dynamics

## TL;DR

This paper introduces a new topological invariant for one-dimensional dynamics based on the bifurcation set, capturing essential dynamical features like entropy and critical point behavior.

## Contribution

It defines and analyzes the bifurcation set as a topological invariant, providing new insights into the structure of transitive and piecewise monotone maps.

## Key findings

- Bifurcation set encodes topological entropy.
- Invariant depends on critical point behavior.
- Provides a geometric-topological perspective on dynamics.

## Abstract

For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their position. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics. We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points.