Periodicity and Indecomposability in Generalized Inverse Limits

TL;DR

This paper investigates how periodicity in upper semicontinuous bonding functions influences the topology of inverse limits, revealing conditions for indecomposable subcontinua and extending classical results in continuum theory.

Contribution

It establishes new links between periodicity in bonding functions and indecomposability of inverse limits, expanding classical continuum theory results.

Findings

Periodic cycles with non-power-of-two periods imply indecomposable subcontinua.

Subcontinua of inverse limits have the full-projection property.

Partial converse results relating periodicity and indecomposability.

Abstract

In this paper, we consider inverse limits of [0,1] using upper semicontinuous set-valued bonding functions with the intermediate value property. Expanding on classical results by Barge and Martin, we explore the relationship between periodicity in the bonding function and the topology of the corresponding inverse limit. In particular, for an inverse limit of a single upper semicontinuous bonding map with the intermediate value property, we provide sufficient conditions for the existence of a periodic cycle with period not a power of two in the bonding function to imply the existence of an indecomposable subcontinuum of the inverse limit. We also give a partial converse. Along the way to these results, we show that subcontinua of these inverse limits have the full-projection property.