Pointwise periodic maps with quantized first integrals

TL;DR

This paper studies specific planar piecewise linear maps that are pointwise periodic, revealing their global dynamics, quantized first integrals, and geometric structure of invariant sets.

Contribution

It introduces new classes of pointwise periodic maps with quantized first integrals and describes their geometric and dynamic properties.

Findings

Existence of discrete, quantized first integrals for these maps.

Invariant level sets are bounded and composed of finite open tiles.

The action of maps on invariant tiles is characterized geometrically.

Abstract

We describe the global dynamics of some pointwise periodic piecewise linear maps in the plane that exhibit interesting dynamic features. For each of these maps we find a first integral. For these integrals the set of values are discrete, thus quantized. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of open tiles of certain regular or uniform tessellations. The action of the maps on each invariant set of tiles is described geometrically.