158 research outputs found
The omega-limit sets of quadratic Julia sets
In this paper we characterize \w-limit sets of dendritic Julia sets for
quadratic maps. We use Baldwin's symbolic representation of these spaces as a
non-Hausdorff itinerary space and prove that quadratic maps with dendritic
Julia sets have shadowing, and also that for all such maps, a closed invariant
set is an \w-limit set of a point if, and only if, it is internally chain
transitive.Comment: 24 page
"Euler Equation Branching"
Some macroeconomic models exhibit a type of global indeterminacy known as Euler equation branching (e.g., the one-sector growth model with a production externality). The dynamics in such models are governed by a differential inclusion. In this paper, we show that in models with Euler equation branching there are multiple equilibria and that the dynamics are chaotic. In particular, we provide sufficient conditions for a dynamical system on the plane with Euler equation branching to be chaotic and show analytically that in a neighborhood of a steady state, these sufficient conditions will typically be satisfied. We also extend the results of Christiano and Harrison (JME, 1999) for the one-sector growth model with a production externality. In a more general setting, we provide necessary and sufficient conditions for Euler equation branching in this model. We show that chaotic and cyclic equilibria are possible and that this behavior is not dependent on the steady state being "locally" determinate or indeterminate.global indeterminacy, Euler equation branching, multiple equilibria, cycles,chaos, increasing returns to scale, externality, regime switching
Expected Utility in Models with Chaos
In this paper, we provide a framework for calculating expected utility in models with chaotic equilibria and consequently a framework for ranking chaos. Suppose that a dynamic economic modelās equilibria correspond to orbits generated by a chaotic dynamical system f : X ! X where X is a compact metric space and f is continuous. The map f could represent the forward dynamics xt+1 = f(xt) or the backward dynamics xt = f(xt+1). If f represents the forward/backward dynamics, the set of equilibria forms a direct/inverse limit space. We use a natural f-invariant measure on X to induce a measure on the direct/inverse limit space and show that this induced measure is a natural Ā¾-invariant measure where Ā¾ is the shift operator. We utilize this framework in the cash-in-advance model of money where f is the backward map to calculate expected utility when equilibria are chaotic.chaos, inverse limits, direct limits, natural invariant measure, cash-in-advance
Shadowing and Expansivity in Sub-Spaces
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.Comment: 18 page
Distributional Chaos in the Baire Space
In this paper we consider the question of distributional chaos on non-compact
metric dynamical systems. We focus on a shift space over a countable alphabet,
the Baire Space. We prove that on the Baire Space subshifts of finite type
exhibit dense distributional chaos and subshifts of bounded type that are
perfect and have a dense set of periodic points also have distributional chaos
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