# Complexity of injective piecewise contracting interval maps

## TL;DR

This paper investigates the complexity growth of itineraries in injective piecewise contracting interval maps, establishing bounds and constructing examples with maximal complexity growth, revealing intricate dynamical structures.

## Contribution

It proves that the complexity function is eventually affine and bounded by the number of discontinuities, and constructs maps achieving this bound with minimal Cantor set dynamics.

## Key findings

- Complexity function is eventually affine.
- Growth rate bounded by number of discontinuities.
- Constructed examples reach the maximum complexity growth.

## Abstract

We study the complexity of the itineraries of injective piecewise contracting maps on the interval. We prove that for any such map the complexity function of any itinerary is eventually affine. We also prove that the growth rate of the complexity is bounded from above by the number $N-1$ of discontinuities of the map. To show that this bound is optimal, we construct piecewise affine contracting maps whose itineraries all have the complexity $(N-1)n +1$. In these examples, the asymptotic dynamics takes place in a minimal Cantor set containing all the discontinuities.