# A spectral decomposition of the attractor of piecewise contracting maps of the interval

## TL;DR

This paper analyzes the long-term behavior of piecewise contracting maps on an interval, revealing a finite decomposition into minimal components like periodic orbits or Cantor sets, and characterizing their origins.

## Contribution

It provides a spectral decomposition of the attractor for non-injective piecewise contracting maps, identifying minimal components and their relation to discontinuities and extrema.

## Key findings

- Decomposition of the support into finite minimal components
- Each component is a periodic orbit or a minimal Cantor set
- Almost every point's omega-limit set is one of these components

## 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 $\omega$-limit set of (almost) every point in the interval is exactly one of these components. Moreover, we show that each component is the $\omega$-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.