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A spectral decomposition of the attractor of piecewise-contracting maps of the interval
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 7 / July 2021
- Published online by Cambridge University Press:
- 05 May 2020, pp. 1940-1960
- Print publication:
- July 2021
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- Article
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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.
