[Paper Review] Ergodic properties of polygonal billiards with strongly contracting reflection laws

Gianluigi Del Magno, João Lopes Dias+2 more|arXiv (Cornell University)|Jan 15, 2015

Mathematical Dynamics and Fractals16 references3 citations

TL;DR

This paper studies ergodic properties of polygonal billiards with strongly contracting reflection laws, where reflection angles are pulled toward the normal. It proves that such billiards have the same number of ergodic SRB measures with identical mixing periods as their corresponding slap maps (orthogonal reflection systems), particularly in polygons without opposing parallel sides, extending known results to a broader class of contracting dynamics.

ABSTRACT

Abstract. We consider convex polygonal billiards with a reflec-tion law that contracts the reflection angle towards the normal. Polygonal billiards with orthogonal reflections are called slap maps. For polygons without parallel sides facing each other, the slap map has a finite number of ergodic absolutely continuous invariant probability measures. We show that any billiard with a strongly contracting reflection law has the same number of ergodic SRB measures with the same mixing periods as the ergodic absolutely continuous invariant probabilities of its corresponding slap map. The case of billiards in regular polygons and triangles is studied in detail. 1.

Motivation & Objective

• To analyze the ergodic invariant measures of polygonal billiards under strongly contracting reflection laws.
• To establish a correspondence between the ergodic SRB measures of contracting billiards and the absolutely continuous invariant probabilities (acip) of their corresponding slap maps.
• To determine the number and mixing periods of ergodic SRB measures in regular polygons and triangles under such dynamics.
• To generalize known results on slap maps to a broader class of reflection laws with strong contraction toward the normal.

Proposed method

• The analysis uses the concept of slap maps—billiards with orthogonal reflections—as a reference system for ergodic invariant measures.
• It applies dynamical systems techniques to compare the structure of invariant measures under strongly contracting reflection laws to those under orthogonal reflections.
• The paper leverages the absence of parallel opposing sides in the polygon to ensure finitely many ergodic acip measures in the slap map case.
• It establishes a one-to-one correspondence between the ergodic SRB measures of the contracting billiard and the ergodic acip measures of the corresponding slap map.
• The proof relies on the contraction property to ensure structural stability and persistence of ergodic components under small perturbations of the reflection law.
• The study of regular polygons and triangles provides concrete settings to verify and illustrate the general correspondence.

Experimental results

Research questions

• RQ1 How do strongly contracting reflection laws affect the number and structure of ergodic SRB measures in polygonal billiards?
• RQ2 What is the relationship between the ergodic SRB measures of a contracting billiard and the ergodic absolutely continuous invariant probabilities of its corresponding slap map?
• RQ3 Do polygons without parallel opposing sides preserve the same number of ergodic measures under contracting dynamics as under orthogonal reflections?
• RQ4 What are the mixing periods of the ergodic SRB measures in regular polygons and triangles under strongly contracting reflection laws?
• RQ5 Can the ergodic properties of contracting billiards be fully characterized by those of their slap map counterparts?

Key findings

• For convex polygonal billiards without parallel sides facing each other, the number of ergodic SRB measures under a strongly contracting reflection law matches the number of ergodic absolutely continuous invariant probabilities in the corresponding slap map.
• The mixing periods of the ergodic SRB measures in the contracting billiard system are identical to those of the corresponding ergodic acip measures in the slap map.
• In regular polygons and triangles, the number of ergodic SRB measures is finite and fully determined by the structure of the corresponding slap map.
• The correspondence between the ergodic measures of the contracting system and the slap map holds due to the strong contraction toward the normal, which stabilizes the dynamics.
• The results extend the known ergodic theory of slap maps to a broader class of reflection laws with strong contraction, preserving key dynamical invariants.