Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics

TL;DR

This paper studies semigroup actions on the interval with specific fixed points and heteroclinic connections, demonstrating that generically they exhibit rapid growth in periodic points and complex universal dynamics under certain conditions.

Contribution

It introduces a model for partially hyperbolic systems showing that generic semigroup actions can have fast periodic point growth and universal dynamics.

Findings

Fast growth of periodic points as a function of period.

Existence of universal dynamics in generic cases.

Conditions on non-linearity and Schwarzian derivative are crucial.

Abstract

We consider semigroup actions on the unit interval generated by strictly increasing $C^r$-maps. We assume that one of the generators has a pair of fixed points, one attracting and one repelling, and a heteroclinic orbit that connects the repeller and attractor, and the other generators form a robust blender, which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller. This is a model setting for partially hyperbolic systems with one central direction. We show that, under additional conditions on the non-linearity and the Schwarzian derivative, the above semigroups exhibit, $C^r$-generically for any r, arbitrarily fast growth of the number of periodic points as a function of the period. We also show that a $C^r$-generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics.