Indecomposable continua and the fixed point property II

Topology Proceedings 4 (1979) pp. 227-234: PRIME ENDS, INDECOMPOSABLE CONTINUA, AND THE FIXED POINT PROPERTY

A Hereditarily Indecomposable Tree-like Continuum without the Fixed Point Property

A hereditarily indecomposable tree-like continuum without the fixed point property is constructed. The example answers a question of Knaster and Bellamy.

Certain Homogeneous Unicoherent Indecomposable Continua

A simple closed curve is the simplest example of a compact, nondegenerate, homogeneous continuum. If a bounded, nondegenerate, homogeneous plane continuum has any local connectedness property, even of the weakest sort, it is known to be a simple closed curve [l, 2, 3].1 The recent discovery of a bounded, nondegenerate, homogenous plane continuum which does not separate the plane [4, 5] has give...

Indecomposable Continua in Exponential Dynamics

In this paper we prove the existence of uncountably many indecomposable continua in the dynamics of complex exponentials of the form Eλ(z) = λe z with λ > 1/e. These continua contain points that share the same itinerary under iteration of Eλ. These itineraries are bounded but consist of blocks of 0’s whose lengths increase, and hence these continua are never periodic.

The fixed point property for arc component preserving mappings of non-metric tree-like continua

The main purpose of this paper is to study the fixed point property of non-metric tree-like continua. Using the inverse systems method, it is proved that if X is a non-metric tree-like continuum and if f : X → X is a mapping which sends each arc component into itself, then f has the fixed point property.