Tiling billiards and Dynnikov’s helicoid. (English) Zbl 1498.37047
Trans. Mosc. Math. Soc. 2021, 133-147 (2021) and Tr. Mosk. Mat. O.-va 82, No. 1, 157-174 (2021).
Summary: Here are two problems. First, understanding the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describing the topology of connected components of plane sections of a centrally symmetric subsurface \(S \subset \mathbb{T}^3\) of genus 3. In this paper we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov’s problem in higher genus seems promising, as we show at the end of this paper.
MSC:
| 37C83 | Dynamical systems with singularities (billiards, etc.) |
| 37B52 | Tiling dynamics |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
Software:
NegSnellReferences:
| [1] | Aichholzer Aichholzer O., Aurenhammer F., Alberts D., G\"artner B. A novel type of skeleton for polygons, J. Univ. Comp. Sci. 1995. Vol.1, No12. P.752-761. 1392429 · Zbl 0943.68171 |
| [2] | ArnouxRauzy1991 Arnoux P., Rauzy G. Repr\'esentation g\'eom\'etrique de suites de complexit\'e \(2n+1\), Bull. Soc. Math. France. 1991. Vol.119, No2. P.199-215. 1116845 · Zbl 0789.28011 |
| [3] | ArnouxStarosta2013 Arnoux P., Starosta \v S. The Rauzy gasket, Further developments in fractals and related fields. New York: Birkh\"auser/Springer, 2013. P.1-23. (Trends Math.) 3184185 |
| [4] | AvilaSkripchenkoHubert Avila A., Hubert P., Skripchenko A. On the Hausdorff dimension of the Rauzy gasket, Bull. Soc. Math. France. 2016. Vol.144, No3. P.539-568. 3558432 · Zbl 1356.37018 |
| [5] | Davis2018 Baird-Smith P., Davis D., Fromm E., Iyer S. Tiling billards on triangle tilings, and interval exchange transformations, Bull. London Math. Soc. 2020. |
| [6] | Baragar1998 Baragar A. The exponent for the Markoff-Hurwitz equations, Pacific J. Math. 1998. Vol.182, No1. P.1-21. 1610610 · Zbl 0892.11009 |
| [7] | DavisNegative2018 Davis D., DiPietro K., Rustad J.T., St.Laurent A. Negative refraction and tiling billiards, Adv. Geom. 2018. Vol.18, No2. P.133-159. 3785417 · Zbl 1393.37048 |
| [8] | DavisHooper2016 Davis D., Hooper W.P. Periodicity and ergodicity in the trihexagonal tiling, Comment. Math. Helv. 2018. Vol.93, No4. P.661-707. 3880224 · Zbl 1411.37044 |
| [9] | DeLeo_2008 DeLeo R., Dynnikov I.A. Geometry of plane sections of the infinite regular skew polyhedron\(\4,6\mid 4\\) , Geom. Dedicata. 2009. Vol.138. P.51-67. 2469987 · Zbl 1165.28006 |
| [10] | Demaine Demaine E.D., O’Rourke J. Geometric folding algorithms. Linkages, origami, polyhedra. Cambridge: Cambridge Univ. Press, 2007. 2354878 · Zbl 1135.52009 |
| [11] | Dynnikov1999TheGO Dynnikov I. The geometry of stability regiones in Novikov’s problem on the semiclassical motion of an electron, Russian Math. Surveys. 1999. Vol.54, No1. P.21-59. 1706843 · Zbl 0935.57040 |
| [12] | DHMPRS2020 Dynnikov I., Hubert P., Mercat P., Paris-Romaskevich O., Skripchenko A. Novikov’s gasket has Lebesgue measure zero. Preprint, 2020. |
| [13] | DHS Dynnikov I., Hubert P., Skripchenko A. Dynamical systems around the Rauzy gasket and their ergodic properties. Preprint, 2020. |
| [14] | DS_band Dynnikov I., Skripchenko A. Symmetric band complexes of thin type and chaotic sections which are not quite chaotic, Trans. MMS. 2015. P.251-269. 3468067 · Zbl 1335.57042 |
| [15] | Fougeron20 Fougeron C. Dynamical properties of simplicial systems and continued fraction algorithms. Preprint, 2020. 2001.01367. |
| [16] | Gamburd Gamburd A., Magee M., Ronan R. An asymptotic formula for integer points on Markoff-Hurwitz varieties, Ann. of Math. (2). 2019. Vol.190, No3. P.751-809. 4024562 · Zbl 1447.11051 |
| [17] | GutierrezMatheus19 Guti\'e R.rrez-Romo, Matheus C. Lower bounds on the dimension of the Rauzy gasket, Bull. Soc. Math. France. 2020. Vol.148, No2. P.321-327. 4124503 · Zbl 1451.37029 |
| [18] | HubertPaRo2019 Hubert P., Paris-Romaskevich O. Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket, Experimental Mathematics. 2019. P.1-30. |
| [19] | Kenyon2018DimersAC Kenyon R., Lam W., Ramassamy S., Russkikh M. Dimers and circle patterns, arXiv: Math. phys., 2018. |
| [20] | KOS2006 Kenyon R., Okounkov A., Sheffield S. Dimers and amoebae, Ann. of Math. (2). 2006. Vol.163, No3. P.1019-1056. 2215138 · Zbl 1154.82007 |
| [21] | MeNo Meester R., Nowicki T. Infinite clusters and critical values in two-dimensional circle percolation, Israel J. Math. 1989. Vol.68, No1. P.63-81. 1035881 · Zbl 0685.60104 |
| [22] | NoDyMa Novikov S.P., De Leo R., Dynnikov I.A., Mal’tsev A. Theory of dynamical systems and transport phenomena in normal metals, J. Exp. Theor. Phys. 2019. Vol.129, No4. P.710-721. |
| [23] | DyNoquasi Novikov S.P., Dynnikov I.A. Topology of quasiperiodic functions on the plane, Russian Math. Surveys. 2005. Vol.60, No1. P.1-26. 2145658 · Zbl 1148.37043 |
| [24] | PaRo2019 Paris-Romaskevich O. Trees and flowers on a billiard table. Preprint, 2019. https://hal.archives-ouvertes.fr/hal-02169195. |
| [25] | Zorich Zorich A. The quasiperiodic structure of level surfaces of a Morse 1-form close to a rational one — a problem of S.P.Novikov, Math. USSR-Izv. 1988. Vol.31, No3. P.635-655. 933967 · Zbl 0694.58002 |
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