The intrinsic metric and geodesics on the Sierpinski gasket SG(3)

The intrinsic metric and geodesics on the Sierpinski gasket SG(3)

We give an explicit expression for the intrinsic metric on the Sierpinski gasket SG(3) (the mod-3 Sierpinskigasket) via code representation of its points. We also investigate the geodesics of SG(3) and determine the number ofgeodesics between two points.

___

  • [1] Barlow MT, Hambly BM. Transition density estimates for brownian motion on scale irregular Sierpinski gaskets. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 1997; 33 (5): 531-557.
  • [2] Barnsley M. Fractals Everywhere. San Diego, CA, USA: Academic Press, 1988.
  • [3] Hilfer R, Blumen A. Renormalisation on Sierpinski-type fractals. Journal of Physics A: Mathematical and General 1984; 17 (10): 537-545. doi: 10.1088/0305-4470/17/10/004
  • [4] Burago D, Burago Y, Ivanov S. A Course in Metric Geometry. USA: AMS, 2001.
  • [5] Chang SC, Chen LC. Number of connected spanning subgraphs on the Sierpinski gasket. Discrete Mathematics and Theoretical Computer Science 2019; 11 (1): 55-78.
  • [6] Cristea LL. A geometric property of the Sierpinski carpet. Quaestiones Mathematicae 2005; 28: 251-262. doi: 10.2989/16073600509486126
  • [7] Cristea LL, Steinsky B. Distances in Sierpinski graphs and on the Sierpinski gasket. Aequationes Mathematicae 2013; 8 (3): 201-219. doi: 10.1007/s00010-013-0197-7
  • [8] Grabner P, Tichy RF. Equidistribution and Brownian motion on the Sierpinski gasket. Monatshefte für Mathematik 1998; 125 (2): 147-164.
  • [9] Gu J, Ye Q, Xi L. Geodesics of higher-dimensional Sierpinski gasket. Fractals 2019; 27 (4): 1950049. doi: 10.1142/S0218348X1950049X
  • [10] Güneri M, Saltan M. Intrinsic metric formulas on some self-similar sets via the code representation. Fractal and Fractional 2019; 3 (1): 1-13. doi: 10.3390/fractalfract3010013
  • [11] Hinz AM, Schief A. The average distance on the Sierpinski gasket. Probability Theory and Related Fields 1990; 87 (1): 129-138.
  • [12] Holter NS, Lakhtakia A, Varadan VK, Varadan VV, Messier R. On a new class of planar fractals: the Pascal- Sierpinski gaskets. Journal of Physics A: Mathematical and General 1986; 19 (9): 1753-1759. doi: 10.1088/0305- 4470/19/9/047
  • [13] Özdemir Y, Saltan M, Demir B. The intrinsic metric on the box fractal. Bulletin of the Iranian Mathematical Society 2019; doi: 10.1007/s41980-018-00197-w
  • [14] Romik D. Shortest paths in the Tower of Hanoi graph and finite automata, SIAM Journal on Discrete Mathematics 2006; 20 (3): 610-622. doi: 10.1137/050628660
  • [15] Saltan M, Özdemir Y, Demir B. An explicit formula of the intrinsic metric on the Sierpinski gasket via code representation. Turkish Journal of Mathematics 2018; 42 (2): 716-725. doi: 10.3906/mat-1702-55
  • [16] Saltan M, Özdemir Y, Demir B. Geodesics of the Sierpinski gasket. Fractals 2018; 26 (3): 1850024. doi: 10.1142/S0218348X1850024X
  • [17] Strichartz RS. Isoperimetric estimates on Sierpinski gasket type fractals. Transactions of the American Mathematical Society 1999; 351 (5): 1705-1752.