Chaos, periodicity, and snakelike continua

Abstract

Abstract. The results of this paper relate the dynamics of a continuous map/of the interval and the topology of the inverse limit space with bonding map /. These inverse limit spaces have been studied by many authors, and are examples of what Bing has called "snakelike continua". Roughly speaking, we show that when the dynamics of / are complicated, the inverse limit space contains indecomposable subcontinua. We also establish a partial converse.

The inverse limits approach to chaos

“…If (I,f) is an inverse limit system such that f is a homeomorphism, then the inverse limit of the inverse system is homeomorphic to an arc, i.e., is homeomorphic to the space [0,1]. Barge and Martin (1985) show that if the dynamics of f are complicated then the inverse limit contains indecomposable subcontinua.…”

“…(Barge and Martin (1985), Theorem 1). Let I = [a, b], f : I → I be continuous, X := lim ← (I, F ) and F : X → X be the induced homeomorphism.…”

“…(Barge and Martin (1985), Theorem 10). Suppose f : I → I is continuous and onto with finitely many turning points.…”

Transitivity, mixing and chaos for a class of set-valued mappings

“…Lemma 3.1 [15] . If f : I → I is a transitive continuous map, then one of the following conditions holds:…”

Intra-orbit separation of dense orbits of interval maps

“…It was shown in [1] that the hypothesis of topological transitivity alone is enough for a map to exhibit chaos (in the sense of Devaney) on compact intervals of R. An easier proof of this result can be found in [14]. Good references of books looking at interval maps are [3,12,13].…”