# The boundary of chaos for piecewise smooth maps and the boundary of positive Hausdorff dimension for survivor sets of open maps

## TL;DR

This paper characterizes the boundary between chaotic and non-chaotic behavior in piecewise smooth maps, linking it to the Hausdorff dimension of survivor sets and identifying two types of codimension one transitions.

## Contribution

It identifies the precise boundary of chaos in piecewise smooth maps and relates it to the boundary of positive Hausdorff dimension for survivor sets, revealing two distinct bifurcation types.

## Key findings

- Boundary of chaos coincides with boundary of positive Hausdorff dimension.
- Two types of codimension one transitions: heteroclinic connections and anharmonic cascades.
- Finite periodic orbits at heteroclinic transition, infinite cascades in anharmonic case.

## Abstract

We describe the boundary of chaos separating regions of parameter space with positive topological entropy from those with zero topological entropy for a class of piecewise smooth maps. This coincides with the boundary of positive Hausdorff dimension for the survivor sets of a class of open maps. There are precisely two types of codimension one transitions across the boundaries. One of these involves heteroclinic connections, and in this case there is a finite number of periodic orbits at the transition point. The other involves an infinite cascade of bifurcations creating infinitely many periodic orbits on the boundary in a sequence called the anharmonic cascade.