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Linear Triangle Dynamics: The Pedal Map and Beyond

Claire Castellano, Corey Manack

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.

Linear Triangle Dynamics: The Pedal Map and Beyond

Abstract

We present a moduli space for similar triangles, then classify triangle maps 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 is mixing, hence ergodic.

Paper Structure

This paper contains 7 sections, 7 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Pedal Map
  3. Moduli space for triangles
  4. Linear maps on $A/G$
  5. Type I triangle maps
  6. Future Research
  7. Acknowledgment.

Key Result

Theorem \oldthetheorem

[Classification of angle transition matrices] Suppose $M:A\to A$ is an ATM. Then there exists $g\in G$ such that $gM$ is either:

Figures (4)

  • Figure 1: The four images display $MA_p$ (shaded in blue) in $A/G$ for various ATM's $M$. From left to right, $M$ is: the identity, the pedal map (Type I), Type II, and Type III
  • Figure 2: The figure on the left shows a reference triangle with first pedal triangle. The figure on the right is the space $A_p$ of interior angles, where the central triangle represents acute triangles and three outer triangles for obtuseness in $\alpha,\beta,\gamma$ respectively.
  • Figure 3: (a) An unfolded part of the moduli space $A/G.$ (b) fundamental domain $D$.
  • Figure 4: The figure on the left is $N^{-1}A_p$ shaded in blue in $A_p$ (white). The figure on the right is the effect of the composition $H_{A,4/5}H_{B,3/4}H_{C,2/3}P$ on $A_p$. The four nested shaded triangles show the composition.

Theorems & Definitions (15)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Remark
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • ...and 5 more