[Paper Review] Families of Periodic Orbits of the Koch Snowflake Fractal Billiard

Michel L. Lapidus, Robert G. Niemeyer|arXiv (Cornell University)|May 4, 2011

Mathematical Dynamics and Fractals35 references3 citations

TL;DR

This paper proposes a framework for defining periodic orbits in the Koch snowflake fractal billiard by taking inverse limits of compatible sequences of periodic orbits from its rational polygonal prefractal approximations. It establishes that certain orbits—particularly those in the π/3 direction—can be characterized as stabilizing or generalized piecewise Fagnano orbits, with their footprints forming self-similar Cantor sets, thereby enabling a rigorous dynamical description on a fractal boundary.

ABSTRACT

We describe the periodic orbits of the prefractal Koch snowflake billiard (the nth inner rational polygonal approximation of the Koch snowflake billiard). In the case of the finite (prefractal) billiard table, we focus on the direction given by an initial angle of pi/3, and define 1) a compatible sequence of piecewise Fagnano orbits, 2) an eventually constant compatible sequence of orbits and 3) a compatible sequence of generalized piecewise Fagnano orbits. In the case of the infinite (fractal) billiard table, we will describe what we call stabilizing periodic orbits of the Koch snowflake fractal billiard. In a sense, we show that it is possible to define billiard dynamics on a Cantor set. In addition, we will show that the inverse limit of the footprints of orbits of the prefractal approximations exists in a specific situation and provide a plausibility argument as to why such an inverse limit of footprints should constitute the footprint of a well-defined periodic orbit of the fractal billiard. Using known results for the inverse limit of a sequence of finite spaces, we deduce that the footprint (i.e., the intersection of the orbit with the boundary) of a piecewise Fagnano orbit is a topological Cantor set and a self-similar Cantor set. We allude to a possible characterization of orbits with an initial direction of pi/3. Such a characterization would allow one to describe an orbit with an initial direction of pi/3 of the Koch snowflake billiard as either a piecewise Fagnano orbit, a stabilizing orbit or a generalized piecewise Fagnano orbit. We then close the paper by discussing several outstanding open problems and conjectures about the Koch snowflake fractal billiard, the associated 'fractal flat surface', and possible connections with the associated fractal drum. In the long-term, the present work may help lay the foundations for a general theory of fractal billiards.

Motivation & Objective

• To define a meaningful notion of billiard dynamics on the Koch snowflake, a nondifferentiable fractal boundary where classical reflection fails.
• To characterize periodic orbits of the Koch snowflake billiard by analyzing compatible sequences of periodic orbits in its rational polygonal prefractal approximations Ω(KSₙ).
• To show that the footprint (intersection with boundary) of such orbits forms a topological and self-similar Cantor set.
• To propose a fractal analog of the Veech dichotomy and explore connections with fractal drums and trace formulae.
• To lay foundational groundwork for a general theory of fractal billiards and their associated geodesic flows on fractal flat surfaces.

Proposed method

• Use inverse limits of Poincaré sections (footprints) of periodic orbits in prefractal billiards Ω(KSₙ) to define orbits in the infinite fractal billiard Ω(KS).
• Analyze the flat surfaces 𝒮(KSₙ) associated with each prefractal billiard, showing they are branched covers of the hexagonal torus 𝒮(KS₀).
• Characterize periodic orbits in direction π/3 using ternary expansions of initial basepoints in Ω(KS₀).
• Define 'compatible sequences' of periodic orbits, including piecewise Fagnano, stabilizing, and generalized piecewise Fagnano types.
• Apply symbolic dynamics and addressing systems to describe the structure of footprints and their inverse limit.
• Leverage known results on inverse limits of finite spaces to prove that the footprint of a piecewise Fagnano orbit is a self-similar Cantor set.

Experimental results

Research questions

• RQ1 Can periodic orbits be rigorously defined in the Koch snowflake billiard despite the absence of a well-defined tangent at the boundary?
• RQ2 What is the geometric and topological structure of the footprint (boundary intersection) of a periodic orbit in the Koch snowflake billiard?
• RQ3 How do periodic orbits in the prefractal approximations Ω(KSₙ) relate to one another across n, and can they form a coherent limit in the fractal case?
• RQ4 Is there a fractal analog of the Veech dichotomy for the Koch snowflake billiard, and if so, under what conditions does it hold?
• RQ5 Can the length spectrum of periodic orbits in the fractal billiard be related to the spectrum of the Laplacian on the associated fractal drum?

Key findings

• The collection of directions for which the billiard flow is closed in Ω(KSₙ) is identical for all n ≥ 0, and matches the set of directions closed in Ω(KS₀).
• The footprint of a piecewise Fagnano orbit in the Koch snowflake billiard is a topological Cantor set, and in fact a self-similar Cantor set due to the fractal symmetry.
• An eventually constant compatible sequence of periodic orbits consists of 𝒞-orbits (stabilizing periodic orbits) for all but finitely many n, and its trivial limit is a well-defined periodic orbit of Ω(KS).
• The inverse limit of the footprints of prefractal orbits exists in specific cases and is proposed as the footprint of a valid periodic orbit in the infinite billiard.
• The Veech group of each prefractal billiard Ω(KSₙ) is commensurate with SL(2,ℤ), supporting the possibility of a fractal analog of the Veech dichotomy.
• The paper provides strong evidence for a complete characterization of orbits with initial direction π/3 as either piecewise Fagnano, stabilizing, or generalized piecewise Fagnano orbits.