Periodic billiard orbits of self-similar Sierpiński carpets

Chen, Joe P.; Niemeyer, Robert G.

doi:10.1016/j.jmaa.2014.03.001

Cited by 12 publications

(12 citation statements)

References 21 publications

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“…The Sierpiński carpets S p and Sierpiński gasket G are easily seen to satisfy Definition 5.15. Moreover, S p contains no non-trivial line segments in directions of irrational slope (see [19,Corollary 4.5] or [12,Theorem 3.4]), and the gasket G clearly contains no nontrivial vertical line segments. Thus, these fractals all satisfy the conditions of Theorem 5.16 and so have Lipschitz dimension at most 1.…”

Section: 34mentioning

confidence: 99%

On the Lipschitz dimension of Cheeger-Kleiner

David¹

2019

Preprint

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In a 2013 paper [9], Cheeger and Kleiner introduced a new type of dimension for metric spaces, the "Lipschitz dimension". We study the dimension-theoretic properties of Lipschitz dimension, including its behavior under Gromov-Hausdorff convergence, its (non-)invariance under various classes of mappings, and its relationship to the Nagata dimension and Cheeger's "analytic dimension". We compute the Lipschitz dimension of various natural spaces, including Carnot groups, snowflakes of Euclidean spaces, metric trees, and Sierpiński carpets. As corollaries, we obtain a short proof of a quasiisometric non-embedding result for Carnot groups and a necessary condition for the existence of non-degenerate Lipschitz maps between certain spaces. GUY C. DAVID 7.2. Lipschitz dimension bounds analytic dimension 30 8. Mapping properties 31 8.1. Lipschitz light mappings 31 8.2. Quasisymmetric and snowflake non-invariance 33 8.3. David-Semmes regularity and non-degenerate Lipschitz maps 33 References 35

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Section: 34mentioning

confidence: 99%

On the Lipschitz dimension of Cheeger-Kleiner

David¹

2019

Preprint

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“…In this paper, we take Ω(F ) to be the T -fractal billiard table shown in Figure 1. In [CheNie1], the fractal billiard table is a self-similar Sierpinski carpet billiard table. In [LapNie1,LapNie2,LapNie3], the fractal billiard table under consideration was the Koch snowflake fractal billiard table.…”

Section: Introductionmentioning

confidence: 99%

“…For the reader's easy reference, we provide in this section various definitions appearing in previous joint works; see [CheNie1,LapNie1,LapNie2,LapNie3,LapNie4]. However, the notation in all of the definitions will reflect the fact that we are discussing the T -fractal billiard table, which will significantly simplify the exposition.…”

Section: Introductionmentioning

confidence: 99%

“…As was alluded to at the beginning of this section, Definitions 2.4 and 2.6 were stated in full generality so that the reader may understand the results in the aforementioned articles, should they refer back to them for examples of periodic orbits and nontrivial paths of other fractal billiard tables (namely, the Koch snowflake fractal billiard and a self-similar Sierpinski carpet fractal billiard table studied, respectively, in , [CheNie1], [LapNie4]). As we see in Figure 5, it may be that two initial conditions are compatible and x 0 0 = x 0 n for all n ≥ 0.…”

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confidence: 99%

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Nontrivial paths and periodic orbits of theT-fractal billiard table

2016

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Abstract. We introduce and prove numerous new results about the orbits of the T -fractal billiard. Specifically, in §3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In §4, we examine the limiting behavior of particular sequences of compatible periodic orbits. Additionally, sufficient conditions for the existence of particular nontrivial paths are given in §4. The proofs of two results stated in [LapNie4] appear here for the first time, as well. In §5, an orbit with an irrational initial direction reaches an elusive point in a way that yields a nontrivial path of finite length, yet, by our convention, constitutes a singular orbit of the fractal billiard table.The existence of such an orbit seems to indicate that the classification of orbits may not be so straightforward. A discussion of our results and directions for future research is then given in §6.

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“…Specifically, § §4.3 and 5.3 contain new results on the prefractal T -fractal billiard Ω(T n ) and the T -fractal billiard Ω(T ) (see [LapNie6]); § §4.4 and 5.4 contain new results for a prefractal Sierpinski carpet billiard Ω(S a,n ) and self-similar Sierpinski carpet billiard Ω(S a ), where a is the single underlying scaling ratio (see [CheNie]). As these sections constitute announcements of new results on the respective prefractal and fractal billiards, we will provide in future papers [CheNie,LapNie4,LapNie5,LapNie6] detailed statements and proofs of the results given therein. Given the nature of the subject of fractal billiards, we will close with a discussion of open problems and possible directions for future work, some of which are to appear in [CheNie] and [LapNie4,LapNie5,LapNie6].…”

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confidence: 99%

The Current State of Fractal Billiards

Lapidus

Niemeyer²

2013

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

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Abstract. If D is a rational polygon, then the associated rational billiard table is given by Ω(D). Such a billiard table is well understood. If F is a closed fractal curve approximated by a sequence of rational polygons, then the corresponding fractal billiard table is denoted by Ω(F ). In this paper, we survey many of the results from [LapNie1-3] for the Koch snowflake fractal billiard Ω(KS) and announce new results on two other fractal billiard tables, namely, the T -fractal billiard table Ω(T ) (see [LapNie6]) and a self-similar Sierpinski carpet billiard table Ω(Sa) (see [CheNie]).We build a general framework within which to analyze what we call a sequence of compatible orbits. Properties of particular sequences of compatible orbits are discussed for each prefractal billiard Ω(KSn), Ω(Tn) and Ω(Sa,n), for n = 0, 1, 2 · · · . In each case, we are able to determine a particular limiting behavior for an appropriately formulated sequence of compatible orbits. Such a limit either constitutes what we call a nontrivial path of a fractal billiard table Ω(F ) or else a periodic orbit of Ω(F ) with finite period. In our examples, F will be either KS, T or Sa. Several of the results and examples discussed in this paper are presented for the first time.We then close with a brief discussion of open problems and directions for further research in the emerging field of fractal billiards.

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Periodic billiard orbits of self-similar Sierpiński carpets

Cited by 12 publications

References 21 publications

On the Lipschitz dimension of Cheeger-Kleiner

On the Lipschitz dimension of Cheeger-Kleiner

Nontrivial paths and periodic orbits of theT-fractal billiard table

The Current State of Fractal Billiards

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