Continuum theory and dynamics problems

A hereditarily indecomposable tree-like continuum without the fixed point property

“…This paper is motivated by the question of whether every tree-like hereditarily indecomposable continuum has the fixed point property. The question was asked by Knaster (Problem 69, [6] and Problem 29, [16]) and Bellamy (page 34, [3]). In this paper we answer the question by proving the following theorem.…”

A Weakly Chainable Tree-Like Continuum without the Fixed Point Property

“…In 1983 Bellamy asked whether every weakly chainable tree-like continuum has the fixed point property [19,Problem 36]; see also [4]. A positive answer to this question seemed likely in view of the theorem in the plane and a classic result by K. Borsuk who, in 1954, proved the fixed point theorem for arcwise-connected tree-like continua [9].…”

A Periodic Point Free Homeomorphism of a Tree-Like Continuum

“…In 1972, T. Ingram [7,Problem 34] (see also [10,Problem 35] and [6,Problem 2]) asked whether each map of a tree-like continuum into itself must have a periodic point. The Bellamy map and other fixed-point free maps on tree-like continua constructed subsequently (see [12], [13], [14] and [11]) have periodic points.…”

“…The Borsuk map is homotopic to the identity. It is not known whether each homotopic to the identity map of a tree-like continuum must have a fixed point (see [10,Problem 27]). Recently, C. Hagopian [9] proved that at least some levels of the homotopy have fixed points.…”