Nonhyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities

Abstract

Abstract. We examine the structure of countable closed invariant sets under a dynamical system on a compact metric space. We are motivated by a desire to understand the possible structures of inhomogeneities in one-dimensional nonhyperbolic sets (inverse limits of finite graphs), particularly when those inhomogeneities form a countable set. Using tools from descriptive set theory we prove a surprising restriction on the topological structure of these invariant sets if the map satisfies a weak repelling or attracting condition. We show that for a family of conceptual models for the Hénon attractor, inverse limits of tent maps, these restrictions characterize the structure of inhomogeneities. We end with several results regarding the collection of parameters that generate such spaces.

On the ω-limit sets of tent maps

“…Furthermore, they have been the subject of many research articles [12,17,28,30], often in relation to their ω-limit sets [6,21,22,23,24]. In this paper we make several important observations about the behaviour of tent maps, allowing us to prove new results about the nature of their limit sets in relation to certain well-known dynamical properties.…”

“…It is known that ω-limit sets are non-empty, closed and invariant (by which we mean that f (ω(x)) = ω(x)). They have been studied extensively, with particular focus on the ω-limit sets of interval maps [1,4,5,6,7,9,10,11,14,15,21,22,23,24].…”

Uncountable ω-limit sets with isolated points

“…We show in [10] that if A is a scattered ω-limit set of a finite-to-one map on a compact metric space, with a weak form of expansivity, then the height of A is a countable ordinal not equal to a limit ordinal or the successor of a limit ordinal, i.e. the empty perfect kernel cannot occur at a limit ordinal.…”

“…The topological structure of the ω-limit set of x is an indication of the complexity of the orbit of x, and as such the topological structure and dynamical features of ω-limit sets is the subject of much study, [1], [2], [4], [7], [8], [10], [11], [14]. Of particular interest is the case where X = [0, 1] and f is a unimodal map with critical point c. In this setting we consider the ω-limit set of the critical point, ω(c).…”

Continuum many tent map inverse limits with homeomorphic postcritical ω-limit sets

“…The collection of folding points of X, denoted by Fd(X), is preserved by any homeomorphism, so understanding the topological structure of this set and how it depends upon the dynamics of f is a necessary step towards a proof of Ingram's conjecture. In [11] we use techniques from descriptive set theory to construct uncountably many (actually ω 1 ) tent map cores (f γ ) γ<ω 1 with critical point c γ and with topologically distinct inverse limit spaces each with ω(c γ ) countable and Fd(lim ← − {[0, 1], f γ }) countable. These spaces are not homeomorphic because the sets of folding points are topologically distinct.…”