Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Asymptotic Properties of Piecewise Contracting Maps   We study the asymptotic dynamics of maps which are piecewise contracting on a
compact space. These maps are Lipschitz continuous, with Lipschitz constant
smaller than one, when restricted to any piece of a finite and dense union of
disjoint open pieces. We focus on the topological and the dynamical properties
of the (global) attractor of the orbits that remain in this union. As a
starting point, we show that the attractor consists of a finite set of periodic
points when it does not intersect the boundary of a contraction piece, which
complements similar results proved for more specific classes of piecewise
contracting maps. Then, we explore the case where the attractor intersects
these boundaries by providing examples that show the rich phenomenology of
these systems. Due to the discontinuities, the asymptotic behaviour is not
always properly represented by the dynamics in the attractor. Hence, we
introduce generalized orbits to describe the asymptotic dynamics and its
recurrence and transitivity properties. Our examples include transitive and
recurrent attractors, that are either finite, countable, or a disjoint union of
a Cantor set and a countable set. We also show that the attractor of a
piecewise contracting map is usually a Lebesgue measure-zero set, and we give
conditions ensuring that it is totally disconnected. Finally, we provide an
example of piecewise contracting map with positive topological entropy and
whose attractor is an interval.

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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: funct-an