# A switched server system semi-conjugate to a minimal interval exchange

## TL;DR

This paper constructs a switched server system whose dynamics are uniquely ergodic and minimal, with all orbits converging to a Cantor attractor, advancing understanding of complex behaviors in such systems.

## Contribution

It provides an explicit example of a switched server system with exotic dynamics, linking it to minimal interval exchange transformations with flips.

## Key findings

- System's orbits converge to a Cantor attractor
- Poincare map semiconjugate to a minimal interval exchange transformation
- Demonstrates exotic, non-periodic asymptotic behavior

## Abstract

Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behavior: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations with flips. The ultimate result is a switched server system whose Poincare map is semiconjugate to a minimal and uniquely ergodic interval exchange transformation with flips.