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A spectral decomposition of the attractor of piecewise-contracting maps of the interval
Published online by Cambridge University Press: 05 May 2020
Abstract
We study the asymptotic dynamics of piecewise-contracting maps defined on a compact interval. For maps that are not necessarily injective, but have a finite number of local extrema and discontinuity points, we prove the existence of a decomposition of the support of the asymptotic dynamics into a finite number of minimal components. Each component is either a periodic orbit or a minimal Cantor set and such that the $\unicode[STIX]{x1D714}$-limit set of (almost) every point in the interval is exactly one of these components. Moreover, we show that each component is the
$\unicode[STIX]{x1D714}$-limit set, or the closure of the orbit, of a one-sided limit of the map at a discontinuity point or at a local extremum.
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- © The Author(s) 2020. Published by Cambridge University Press
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