Brian E. Raines

A characterization of ω-limit sets in shift spaces

A set 3 is internally chain transitive if for any x; y23 and > 0 there is an -pseudo-orbit in3 between x and y. In this paper we characterize all!-limit sets in shifts of finite type by showing that, if3 is a closed, strongly shift-invariant subset of a shift of finite type, X , then there is a point z2 X with!.z/D3 if and only if3 is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet B is the!-limit set of some point in the full shift space over B. We use similar techniques to prove that, for a tent map f , a closed, strongly f -invariant, internally chain transitive subset of the interval is the!-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space ZG (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the!-limit set of any point in ZG.

Fixed points imply chaos for a class of differential inclusions that arise in economic models

We consider multi-valued dynamical systems with continuous time of the form ẋ ∈ F (x), where F (x) is a set-valued function. Such models have been studied recently in mathematical economics. We provide a definition for chaos, ω-chaos and topological entropy for these differential inclusions that is in terms of the natural R-action on the space of all solutions of the model. By considering this more complicated topological space and its R-action we show that chaos is the ‘typical’ behavior in these models by showing that near any hyperbolic fixed point there is a region where the system is chaotic, ω-chaotic, and has infinite topological entropy.

Basins of measures on inverse limit spaces for the induced homeomorphism

Let fV X! X be continuous and onto, where X is a compact metric space. Let YVD lim .X; f/ be the inverse limit and FV Y! Y the induced homeomorphism. Suppose that is an f -invariant measure, and let m be the measure induced on Y by .;; : : :/ . We show that B is a basin of if and only if 1 1 .B/ is a basin of m. From this it follows that if is an SRB measure for f on X , then the induced measure m on Y is an inverse- limit SRB measure for F. Conversely, if m is an inverse-limit SRB measure for F on Y , then the induced measure on X is an SRB measure for f .

Shadowing and -limit sets of circular Julia sets

We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval, in which context some of our results can be extended.

IMPLICIT EQUILIBRIUM DYNAMICS

We discuss the problem known in economics as backward dynamics occurring in models of perfect foresight, intertemporal equilibrium described mathematically by implicit difference equations. In a previously published paper [ Journal of Economic Dynamics and Control 31 (2007), 1633–1671], we showed that by means of certain mathematical methods and results known as inverse limits theory it is possible to establish a correspondence between the backward dynamics of a noninvertible map and the forward dynamics of a related invertible map acting on an appropriately defined space of sequences, each of whose elements corresponds to an intertemporal equilibrium. We also proved the existence of different types of topological attractors for one-dimensional models of overlapping generations. In this paper, we provide an extension of those results, constructing a Lebesgue-like probability measure on spaces of infinite sequences that allows us to distinguish typical from exceptional dynamical behaviors in a measure–theoretical sense, thus proving that all the topological attractors considered in MR07 are also metric attractors. We incidentally also prove that the existence of chaos (in the Devaney–Touhey sense) backward in time implies (and is implied by) chaos forward in time.