Linear Triangle Dynamics: The Pedal Map and Beyond

TL;DR

This paper introduces a moduli space for similar triangles, classifies linear triangle maps including the pedal map, and proves these maps are mixing and ergodic using Markov partitions.

Contribution

It provides a new classification of linear triangle maps and demonstrates their dynamical properties, extending the understanding of the pedal map within a broader framework.

Findings

Linear triangle maps admit Markov partitions

All such maps are mixing and ergodic

The pedal map is a special case within this classification

Abstract

We present a moduli space for similar triangles, then classify triangle maps $f$ that arise from linear maps on this space, with the well-studied pedal map as a special case. Each linear triangle map admits a Markov partition, showing that $f$ is mixing, hence ergodic.