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We give a rigorous definition of the T-fractal translation surface, and describe some its basic geometric and dynamical properties. In particular, we study the singularities attached to the surface by its metric completion and show there exists a Cantor set of "elusive singularities." We show these elusive singularities can be thought of as a generalization of the wild singularities introduced by Bowman and Valdez. In particular, we show that every elusive singularities has an infinite discrete set of rotational components., Comment: 30 pages, 18 figures. Comments welcome

The works of [Cha-DunAlvInoNieCarFieLaw,Cha-Dun] describe upward sweeps in populations of city-states and attempt to characterize such phenomenon. The model proposed in both [TurKor,Tur] describes how the population, state resources and internal conflict influence each other over time. We show that one can obtain an upward sweep in the population by altering particular parameters of the system of differential equations constituting the model given in [TurKor,Tur]. Moreover, we show that such a system has a unstable critical point and propose an approach for determining bifurcation points in the parameter space for the model., Comment: 20 pages, 13 figures. Contains Matlab code for those interested in adapting and/or extending the model. Comments are welcome

We introduce and prove numerous new results about the orbits of the $T$-fractal billiard. Specifically, in Section 3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In Section 4, we examine the limiting behavior of particular sequences of compatible periodic orbits and, more interesting, in Section 5, the limiting behavior of a particular sequence of compatible singular orbits. The latter seems to indicate that the classification of orbits may not be so straightforward. Additionally, sufficient conditions for the existence of particular nontrivial paths is given in Section 4. The proofs of two results stated in [LapNie4] appear here for the first time, as well. A discussion of our results and directions for future research is then given in Section 6., Comment: 20 Figures, 35 pages, two results from arXiv:1210.0282 are generalized and proved in this article. Many new results appear here. Comments welcome. Version 3 contains minor grammatical changes and the presentation of some results has greatly improved. To appear in the journal Nonlinearity

A formal definition of a (mathematical) polygonal Andreev billiard and a construction of an equivalence relation that captures the dynamics described in physical toy model of Andreev reflection are given. The continuous flow and discrete flow on the respective phase spaces. It is then shown that the continuous flow preserves the absolute value of the volume element $dx\wedge dy\wedge d\theta$ and the billiard (collision) map preserves the measure $\cos \phi dr d\phi$, respectively. One can then characterize the dynamics of a rational polygonal Andreev billiard table. Finally, a discussions of the effect of a fractal perturbation of the toy model of a rectangular nanowire lying upon a superconducting medium is given., Comment: This is a preliminary report and open for comments and suggestions. 17 pages, 15 figures. Makes reference to arXiv:1503.08492

We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table. Based on a refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in a self-similar Sierpinski carpet, we construct what is called an eventually constant sequence of compatible periodic orbits of prefractal Sierpinski carpet billiard tables. The trivial limit of this sequence then constitutes a periodic orbit of a self-similar Sierpinski carpet billiard table. We also determine the corresponding translation surface for each prefractal billiard table, and show that the genera of a sequence of translation surfaces increase without bound. Various open questions and possible directions for future research are offered., Comment: 22 Figures

If D is a rational polygon, then the associated rational billiard table is given by \Omega(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 \Omega(F). In this paper, we survey many of the results from [LapNie1-3] for the Koch snowflake fractal billiard \Omega(KS) and announce new results on two other fractal billiard tables, namely, the T-fractal billiard table \Omega(T) (see [LapNie6]) and a self-similar Sierpinski carpet billiard table \Omega(S_a) (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 \Omega(KS_n), \Omega(T_n) and \Omega(S_a,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 \Omega(F) or else a periodic orbit of \Omega(F) with finite period. In our examples, F will be either KS, T or S_a. 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., Comment: 26 figures. his article is a survey of resent results, in addition to newly announced results, proofs for which are to appear in forthcoming papers; arXiv admin note: text overlap with arXiv:1204.3133

The Koch snowflake KS is a nowhere differentiable curve. The billiard table Omega(KS) with boundary KS is, a priori, not well defined. That is, one cannot a priori determine the minimal path traversed by a billiard ball subject to a collision in the boundary of the table. It is this problem which makes Omega(KS) such an interesting, yet difficult, table to analyze. In this paper, we approach this problem by approximating (from the inside) Omega(KS) by well-defined (prefractal) rational polygonal billiard tables Omega(KS_n). We first show that the flat surface S(KS_n) determined from the rational billiard Omega(KS_n) is a branched cover of the singly punctured hexagonal torus. Such a result, when combined with the results of [Gut2], allows us to define a sequence of compatible orbits of prefractal billiards. We define a hybrid orbit of a prefractal billiard Omega(KS_n) and show that every dense orbit of a prefractal billiard is a dense hybrid orbit of Omega(KS_n). This result is key in obtaining a topological dichotomy for a sequence of compatible orbits. Furthermore, we determine a sufficient condition for a sequence of compatible orbits to be a sequence of compatible periodic hybrid orbits. We then examine the limiting behavior of a sequence of compatible periodic hybrid orbits. We show that the trivial limit of particular (eventually) constant sequences of compatible hybrid orbits constitutes an orbit of Omega(KS). In addition, we show that the union of two suitably chosen nontrivial polygonal paths connects two elusive limit points of the Koch snowflake. Finally, we discuss how it may be possible for our results to be generalized to other fractal billiard tables and how understanding the structures of the Veech groups of the prefractal billiards may help in determining `fractal flat surfaces' naturally associated with the billiard flows., Comment: 21 pages, 15 figures

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., Comment: This paper contains an index of notation, a table of contents and 31 figures; it is 63 pages in length

In this paper, we attempt to define and understand the orbits of the Koch snowflake fractal billiard $KS$. This is a priori a very difficult problem because $\partial(KS)$, the snowflake curve boundary of $KS$, is nowhere differentiable, making it impossible to apply the usual law of reflection at any point of the boundary of the billiard table. Consequently, we view the prefractal billiards $KS_n$ (naturally approximating $KS$ from the inside) as rational polygonal billiards and examine the corresponding flat surfaces of $KS_n$, denoted by $\mathcal{S}_{KS_n}$. In order to develop a clearer picture of what may possibly be happening on the billiard $KS$, we simulate billiard trajectories on $KS_n$ (at first, for a fixed $n\geq 0$). Such computer experiments provide us with a wealth of questions and lead us to formulate conjectures about the existence and the geometric properties of periodic orbits of $KS$ and detail a possible plan on how to prove such conjectures., Comment: 26 pages, color figures (For crisper figures, please contact the second author)