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arXiv · 2508.21429

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

Paul Glendinning·Clément Hege
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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.

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Paul Glendinning, Clément Hege. 2025-08-29. The boundary of chaos for piecewise smooth maps and the boundary of positive Hausdorff dimension for survivor sets of open maps. https://arxiv.org/abs/2508.21429

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