Omega-limit sets for spiral maps

Bruce Kitchens,Michał Misiurewicz

doi:10.3934/dcds.2010.27.787

Abstract

We investigate a class of homeomorphisms of a cylinder, with alltrajectories convergent to the cylinder base and one fixed point inthe base. Let A be a nonempty finite or countable family ofsets, each of which can be a priori an $\omega$-limit set. Then thereis a homeomorphism from our class, for which A is the family ofall $\omega$-limit sets.

Highlights

For a simple map of a triangle, defined and investigated 50 years ago by P
Maybe the family of all ω-limit sets is similar for a large class of homeomorphisms with very similar dynamics? In this paper we show that this is not the case
The main result of the paper is the existence of regular spiral cylinder maps for which the set of all ω-limit sets of points from A ∖ B is equal to a prescribed nonempty finite or countable family of admissible sets (Theorem 3.1)

Summary

Introduction

For a simple map of a triangle, defined and investigated 50 years ago by P. It makes sense to remove a small disk centered at the central fixed point and conjugate the system to a homeomorphism of a cylinder This motivates the following definition of a class of cylinder homeomorphisms with the dynamics very similar to the one of the Stein-Ulam Spiral map. It is an open question whether the Stein-Ulam Spiral map is (after removing a neighborhood of the central fixed point and dividing by symmetries) conjugate to a regular cylinder spiral map. Our counterexample is in a sense stronger than the triangular one – because our map is invertible – and stronger than the trivial one – because for our map all ω-limit sets (except the singleton of the central fixed point) are contained in the boundary of the circle, so the system is almost one-dimensional. In the fibers over the terms of our sequence the whole interval will be an ω-limit set, while in the limit fiber, over 0, the only limit sets will be singletons

Nonautonomous systems on the circle

Findings

Cylinder maps