Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

Daniel J. Kelleher; Hugo Panzo; Antoni Brzoska; Alexander Teplyaev

doi:10.3934/dcdss.2019002

Previous Article
Asymptotics for quasilinear obstacle problems in bad domains
DCDS-S Home
This Issue
Next Article
Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d-sets

February 2019, 12(1): 27-42. doi: 10.3934/dcdss.2019002

Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

Daniel J. Kelleher ^1, , Hugo Panzo ^2, , Antoni Brzoska ^2, and Alexander Teplyaev ^2,

1.	Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA 01075, USA
2.	Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

URL: https://djkelleher.wordpress.com/
URL: http://www.math.uconn.edu/~panzo/
URL: http://www.math.uconn.edu/~teplyaev/

Received November 2016 Revised October 2017 Published July 2018

Fund Project: Research supported in part by NSF grants DMS 1106982, 1262929, 1613025.

We prove Barlow-Bass type resistance estimates for two random walks associated with repeated barycentric subdivisions of a triangle. If the random walk jumps between the centers of triangles in the subdivision that have common sides, the resistance scales as a power of a constant $ρ$ which is theoretically estimated to be in the interval $5/4≤ρ≤3/2$, with a numerical estimate $ρ≈1.306$. This corresponds to the theoretical estimate of spectral dimension $d_S$ between 1.63 and 1.77, with a numerical estimate $d_S≈1.74$. On the other hand, if the random walk jumps between the corners of triangles in the subdivision, then the resistance scales as a power of a constant $ρ^T = 1/ρ$, which is theoretically estimated to be in the interval $2/3≤ρ^T≤4/5$. This corresponds to the spectral dimension between 2.28 and 2.38. The difference between $ρ$ and $ρ^T$ implies that the the limiting behavior of random walks on the repeated barycentric subdivisions is more delicate than on the generalized Sierpinski Carpets, and suggests interesting possibilities for further research, including possible non-uniqueness of self-similar Dirichlet forms.

Keywords: Repeated barycentric subdivision, random walk, Sierpinski carpet, Strichartz hexacarpet, Barlow–Bass resistance estimate, spectral dimension, self-similar Dirichlet form.

Mathematics Subject Classification: Primary: 60J35, 81Q35, 28A80; Secondary: 31C25, 31E05, 35K08.

Citation: Daniel J. Kelleher, Hugo Panzo, Antoni Brzoska, Alexander Teplyaev. Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (1) : 27-42. doi: 10.3934/dcdss.2019002

References:

[1]	L. Ambrosio, M. Erbar and G. Savaré, Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces, Nonlinear Anal., 137 (2016), 77-134. doi: 10.1016/j.na.2015.12.006. Google Scholar
[2]	M. T. Barlow, Analysis on the Sierpinski carpet, in Analysis and Geometry of Metric Measure Spaces, vol. 56 of CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, 2013, 27–53. Google Scholar
[3]	M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. London Ser. A, 431 (1990), 345-360. doi: 10.1098/rspa.1990.0135. Google Scholar
[4]	M. T. Barlow, R. F. Bass and J. D. Sherwood, Resistance and spectral dimension of Sierpiński carpets, J. Phys. A, 23 (1990), L253-L258. doi: 10.1088/0305-4470/23/6/004. Google Scholar
[5]	M. Barlow, Diffusions on fractals, in Lectures on Probability Theory and Statistics (SaintFlour, 1995), vol. 1690 of Lecture Notes in Math., Springer, Berlin, 1998, 1–121. doi: 10.1007/BFb0092537. Google Scholar
[6]	M. Barlow and R. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math., 51 (1999), 673-744. doi: 10.4153/CJM-1999-031-4. Google Scholar
[7]	M. Barlow, R. F. Bass, T. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets, J. Eur. Math. Soc. (JEMS), 12 (2010), 655-701. Google Scholar
[8]	L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets, in Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 25–118. Google Scholar
[9]	R. Bass, Diffusions on the Sierpinski carpet, in Trends in Probability and Related Analysis (Taipei, 1996), World Sci. Publ., River Edge, NJ, 1997, 1–34. Google Scholar
[10]	F. Baudoin and D. J. Kelleher, Differential one-forms on dirichlet spaces and bakry-emery estimates on metric graphs, arXiv: 1604.02520, Transactions of the AMS, to appear. doi: 10.1090/tran/7362. Google Scholar
[11]	F. Bauer, M. Keller and R. K. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, J. Eur. Math. Soc. (JEMS), 17 (2015), 259-271. doi: 10.4171/JEMS/503. Google Scholar
[12]	M. Begue, D. Kelleher, A. Nelson, H. Panzo, R. Pellico and A. Teplyaev, Random walks on barycentric subdivisions and the Strichartz hexacarpet, Exp. Math., 21 (2012), 402-417. doi: 10.1080/10586458.2012.715542. Google Scholar
[13]	R. Bell, C.-W. Ho and R. S. Strichartz, Energy measures of harmonic functions on the Sierpiński gasket, Indiana Univ. Math. J., 63 (2014), 831-868. doi: 10.1512/iumj.2014.63.5256. Google Scholar
[14]	J. Bello, Y. Li and R. S. Strichartz, Hodge–de Rham theory of K-forms on carpet type fractals, in Excursions in Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 3 (2015), 23–62. Google Scholar
[15]	N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, vol. 14 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1991. doi: 10.1515/9783110858389. Google Scholar
[16]	Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, vol. 35 of London Mathematical Society Monographs Series, Princeton Univ. Press, 2012. Google Scholar
[17]	P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76. doi: 10.1137/S0036144598338446. Google Scholar
[18]	P. Diaconis and C. McMullen, Barycentric Subdivision Unpublished, 2008. Google Scholar
[19]	P. Diaconis and L. Miclo, On barycentric partitions, with simulations, https://hal.archives-ouvertes.fr/hal-00353842. Google Scholar
[20]	P. Diaconis and L. Miclo, On barycentric subdivision, Combin. Probab. Comput., 20 (2011), 213-237. doi: 10.1017/S0963548310000441. Google Scholar
[21]	P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, vol. 22 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1984. Google Scholar
[22]	M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, vol. 19 of de Gruyter Studies in Mathematics, extended edition, Walter de Gruyter & Co., Berlin, 2011. Google Scholar
[23]	R. Grigorchuk and V. Nekrashevych, Self-similar groups, operator algebras and Schur complement, J. Mod. Dyn., 1 (2007), 323-370. doi: 10.3934/jmd.2007.1.323. Google Scholar
[24]	A. Grigor'yan and J. Hu, Heat kernels and Green functions on metric measure spaces, Canad. J. Math., 66 (2014), 641-699. doi: 10.4153/CJM-2012-061-5. Google Scholar
[25]	A. Grigor'yan and A. Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab., 40 (2012), 1212-1284. doi: 10.1214/11-AOP645. Google Scholar
[26]	B. M. Hambly, V. Metz and A. Teplyaev, Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc. (2), 74 (2006), 93-112. doi: 10.1112/S002461070602312X. Google Scholar
[27]	K. E. Hare, B. A. Steinhurst, A. Teplyaev and D. Zhou, Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals, Math. Res. Lett., 19 (2012), 537-553. doi: 10.4310/MRL.2012.v19.n3.a3. Google Scholar
[28]	A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Google Scholar
[29]	J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, vol. 27 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2015, An approach based on upper gradients. doi: 10.1017/CBO9781316135914. Google Scholar
[30]	M. Hinz, D. Kelleher and A. Teplyaev, Measures and Dirichlet forms under the Gelfand transform, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 408 (2012), 303-322, 329–330. Google Scholar
[31]	M. Hinz and A. Teplyaev, Closability, regularity, and approximation by graphs for separable bilinear forms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 441 (2015), 299-317. Google Scholar
[32]	M. Hinz, D. J. Kelleher and A. Teplyaev, Metrics and spectral triples for Dirichlet and resistance forms, J. Noncommut. Geom., 9 (2015), 359-390. doi: 10.4171/JNCG/195. Google Scholar
[33]	M. Hinz, M. R. Lacia, A. Teplyaev and P. Vernole, Fractal snowflake domain diffusion with boundary and interior drifts, J. Math. Anal. Appl., 457 (2018), 672–693, arXiv: 1605.06785. doi: 10.1016/j.jmaa.2017.07.065. Google Scholar
[34]	M. Hinz, M. Röckner and A. Teplyaev, Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces, Stochastic Process. Appl., 123 (2013), 4373-4406. doi: 10.1016/j.spa.2013.06.009. Google Scholar
[35]	M. Hinz and A. Teplyaev, Dirac and magnetic Schrödinger operators on fractals, J. Funct. Anal., 265 (2013), 2830-2854. doi: 10.1016/j.jfa.2013.07.021. Google Scholar
[36]	M. Ionescu, L. Rogers and A. Teplyaev, Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169. doi: 10.1016/j.jfa.2012.05.021. Google Scholar
[37]	V. A. Kaimanovich, "Münchhausen trick" and amenability of self-similar groups, Internat. J. Algebra Comput., 15 (2005), 907-937. doi: 10.1142/S0218196705002694. Google Scholar
[38]	N. Kajino, Heat kernel asymptotics for the measurable Riemannian structure on the Sierpinski gasket, Potential Anal., 36 (2012), 67-115. doi: 10.1007/s11118-011-9221-5. Google Scholar
[39]	N. Kajino, Analysis and geometry of the measurable Riemannian structure on the Sierpiński gasket, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. I. Fractals in Pure Mathematics, vol. 600 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2013, 91–133. doi: 10.1090/conm/600/11932. Google Scholar
[40]	C. J. Kauffman, R. M. Kesler, A. G. Parshall, E. A. Stamey and B. A. Steinhurst, Quantum mechanics on Laakso spaces, J. Math. Phys., 53(2012), 042102, 18pp. doi: 10.1063/1.3702099. Google Scholar
[41]	D. J. Kelleher, B. A. Steinhurst and C. -M. M. Wong, From self-similar structures to selfsimilar groups, Internat. J. Algebra Comput., 22 (2012), 1250056, 16pp. doi: 10.1142/S0218196712500567. Google Scholar
[42]	M. Keller, D. Lenz and R. K. Wojciechowski, Volume growth, spectrum and stochastic completeness of infinite graphs, Math. Z., 274 (2013), 905-932. doi: 10.1007/s00209-012-1101-1. Google Scholar
[43]	J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755. doi: 10.2307/2154402. Google Scholar
[44]	J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. Google Scholar
[45]	J. Kigami, Local Nash inequality and inhomogeneity of heat kernels, Proc. London Math. Soc. (3), 89 (2004), 525-544. doi: 10.1112/S0024611504014807. Google Scholar
[46]	J. Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc., 199 (2009), ⅷ+94pp. doi: 10.1090/memo/0932. Google Scholar
[47]	J. Kigami, Quasisymmetric modification of metrics on self-similar sets, in Geometry and Analysis of Fractals, vol. 88 of Springer Proc. Math. Stat., Springer, Heidelberg, 2014,253–282. doi: 10.1007/978-3-662-43920-3_9. Google Scholar
[48]	O. Knill, The graph spectrum of barycentric refinements, arXiv: 1508.02027. Google Scholar
[49]	O. Knill, Universality for Barycentric subdivision, arXiv: 1509.06092. Google Scholar
[50]	S. Kusuoka and X. Y. Zhou, Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Related Fields, 93 (1992), 169-196. doi: 10.1007/BF01195228. Google Scholar
[51]	M. Lapidus and J. Sarhad, Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets, J. Noncommut. Geom., 8 (2014), 947-985. doi: 10.4171/JNCG/174. Google Scholar
[52]	P. Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2), 124 (1986), 1-21. doi: 10.2307/1971385. Google Scholar
[53]	P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201. doi: 10.1007/BF02399203. Google Scholar
[54]	T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990), ⅳ+128pp. doi: 10.1090/memo/0420. Google Scholar
[55]	D. Lougee and B. Steinhurst, Bond percolation on a non-P. C. F. Sierpiński gasket, iterated barycentric subdivision of a triangle, and hexacarpet, Fractals, 24 (2016), 1650023, 12pp. doi: 10.1142/S0218348X16500237. Google Scholar
[56]	R. Lyons and Y. Peres, Probability on Trees and Networks, vol. 42 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, New York, 2016, Available at http://pages.iu.edu/~rdlyons/. doi: 10.1017/9781316672815. Google Scholar
[57]	I. McGillivray, Resistance in higher-dimensional Sierpiński carpets, Potential Anal., 16 (2002), 289-303. doi: 10.1023/A:1014035414658. Google Scholar
[58]	D. Molitor, N. Ott and R. Strichartz, Using Peano curves to construct Laplacians on fractals, Fractals, 23 (2015), 1550048, 29pp. doi: 10.1142/S0218348X15500486. Google Scholar
[59]	V. Nekrashevych, Self-similar Groups, vol. 117 of Mathematical Surveys and Monographs, Amer. Math. Soc., 2005. doi: 10.1090/surv/117. Google Scholar
[60]	V. Nekrashevych and A. Teplyaev, Groups and analysis on fractals, in Analysis on Graphs and Its Applications, vol. 77 of Proc. Sympos. Pure Math., Amer. Math. Soc., 2008, 143–180. doi: 10.1090/pspum/077/2459868. Google Scholar
[61]	L. Rogers and A. Teplyaev, Laplacians on the basilica Julia sets, Commun. Pure Appl. Anal., 9 (2010), 211-231. doi: 10.3934/cpaa.2010.9.211. Google Scholar
[62]	B. Steinhurst, Uniqueness of locally symmetric Brownian motion on Laakso spaces, Potential Anal., 38 (2013), 281-298. doi: 10.1007/s11118-012-9273-1. Google Scholar
[63]	R. S. Strichartz, Differential Equations on Fractals. A Tutorial, Princeton Univ. Press, 2006. Google Scholar
[64]	A. Telcs and V. Vespri, Resolvent metric and the heat kernel estimate for random walks, Stochastic Process. Appl., 124 (2014), 3965-3985. doi: 10.1016/j.spa.2014.07.012. Google Scholar
[65]	A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure, Canad. J. Math., 60 (2008), 457-480. doi: 10.4153/CJM-2008-022-3. Google Scholar
[66]	S. Volkov, Random geometric subdivisions, Random Structures Algorithms, 43 (2013), 115-130. doi: 10.1002/rsa.20454. Google Scholar

show all references

URL: https://djkelleher.wordpress.com/
URL: http://www.math.uconn.edu/~panzo/
URL: http://www.math.uconn.edu/~teplyaev/

References:

[1]	L. Ambrosio, M. Erbar and G. Savaré, Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces, Nonlinear Anal., 137 (2016), 77-134. doi: 10.1016/j.na.2015.12.006. Google Scholar
[2]	M. T. Barlow, Analysis on the Sierpinski carpet, in Analysis and Geometry of Metric Measure Spaces, vol. 56 of CRM Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, 2013, 27–53. Google Scholar
[3]	M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. London Ser. A, 431 (1990), 345-360. doi: 10.1098/rspa.1990.0135. Google Scholar
[4]	M. T. Barlow, R. F. Bass and J. D. Sherwood, Resistance and spectral dimension of Sierpiński carpets, J. Phys. A, 23 (1990), L253-L258. doi: 10.1088/0305-4470/23/6/004. Google Scholar
[5]	M. Barlow, Diffusions on fractals, in Lectures on Probability Theory and Statistics (SaintFlour, 1995), vol. 1690 of Lecture Notes in Math., Springer, Berlin, 1998, 1–121. doi: 10.1007/BFb0092537. Google Scholar
[6]	M. Barlow and R. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math., 51 (1999), 673-744. doi: 10.4153/CJM-1999-031-4. Google Scholar
[7]	M. Barlow, R. F. Bass, T. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets, J. Eur. Math. Soc. (JEMS), 12 (2010), 655-701. Google Scholar
[8]	L. Bartholdi, R. Grigorchuk and V. Nekrashevych, From fractal groups to fractal sets, in Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 25–118. Google Scholar
[9]	R. Bass, Diffusions on the Sierpinski carpet, in Trends in Probability and Related Analysis (Taipei, 1996), World Sci. Publ., River Edge, NJ, 1997, 1–34. Google Scholar
[10]	F. Baudoin and D. J. Kelleher, Differential one-forms on dirichlet spaces and bakry-emery estimates on metric graphs, arXiv: 1604.02520, Transactions of the AMS, to appear. doi: 10.1090/tran/7362. Google Scholar
[11]	F. Bauer, M. Keller and R. K. Wojciechowski, Cheeger inequalities for unbounded graph Laplacians, J. Eur. Math. Soc. (JEMS), 17 (2015), 259-271. doi: 10.4171/JEMS/503. Google Scholar
[12]	M. Begue, D. Kelleher, A. Nelson, H. Panzo, R. Pellico and A. Teplyaev, Random walks on barycentric subdivisions and the Strichartz hexacarpet, Exp. Math., 21 (2012), 402-417. doi: 10.1080/10586458.2012.715542. Google Scholar
[13]	R. Bell, C.-W. Ho and R. S. Strichartz, Energy measures of harmonic functions on the Sierpiński gasket, Indiana Univ. Math. J., 63 (2014), 831-868. doi: 10.1512/iumj.2014.63.5256. Google Scholar
[14]	J. Bello, Y. Li and R. S. Strichartz, Hodge–de Rham theory of K-forms on carpet type fractals, in Excursions in Harmonic Analysis, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 3 (2015), 23–62. Google Scholar
[15]	N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, vol. 14 of de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1991. doi: 10.1515/9783110858389. Google Scholar
[16]	Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, vol. 35 of London Mathematical Society Monographs Series, Princeton Univ. Press, 2012. Google Scholar
[17]	P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76. doi: 10.1137/S0036144598338446. Google Scholar
[18]	P. Diaconis and C. McMullen, Barycentric Subdivision Unpublished, 2008. Google Scholar
[19]	P. Diaconis and L. Miclo, On barycentric partitions, with simulations, https://hal.archives-ouvertes.fr/hal-00353842. Google Scholar
[20]	P. Diaconis and L. Miclo, On barycentric subdivision, Combin. Probab. Comput., 20 (2011), 213-237. doi: 10.1017/S0963548310000441. Google Scholar
[21]	P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, vol. 22 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1984. Google Scholar
[22]	M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, vol. 19 of de Gruyter Studies in Mathematics, extended edition, Walter de Gruyter & Co., Berlin, 2011. Google Scholar
[23]	R. Grigorchuk and V. Nekrashevych, Self-similar groups, operator algebras and Schur complement, J. Mod. Dyn., 1 (2007), 323-370. doi: 10.3934/jmd.2007.1.323. Google Scholar
[24]	A. Grigor'yan and J. Hu, Heat kernels and Green functions on metric measure spaces, Canad. J. Math., 66 (2014), 641-699. doi: 10.4153/CJM-2012-061-5. Google Scholar
[25]	A. Grigor'yan and A. Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab., 40 (2012), 1212-1284. doi: 10.1214/11-AOP645. Google Scholar
[26]	B. M. Hambly, V. Metz and A. Teplyaev, Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc. (2), 74 (2006), 93-112. doi: 10.1112/S002461070602312X. Google Scholar
[27]	K. E. Hare, B. A. Steinhurst, A. Teplyaev and D. Zhou, Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals, Math. Res. Lett., 19 (2012), 537-553. doi: 10.4310/MRL.2012.v19.n3.a3. Google Scholar
[28]	A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. Google Scholar
[29]	J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, vol. 27 of New Mathematical Monographs, Cambridge University Press, Cambridge, 2015, An approach based on upper gradients. doi: 10.1017/CBO9781316135914. Google Scholar
[30]	M. Hinz, D. Kelleher and A. Teplyaev, Measures and Dirichlet forms under the Gelfand transform, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 408 (2012), 303-322, 329–330. Google Scholar
[31]	M. Hinz and A. Teplyaev, Closability, regularity, and approximation by graphs for separable bilinear forms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 441 (2015), 299-317. Google Scholar
[32]	M. Hinz, D. J. Kelleher and A. Teplyaev, Metrics and spectral triples for Dirichlet and resistance forms, J. Noncommut. Geom., 9 (2015), 359-390. doi: 10.4171/JNCG/195. Google Scholar
[33]	M. Hinz, M. R. Lacia, A. Teplyaev and P. Vernole, Fractal snowflake domain diffusion with boundary and interior drifts, J. Math. Anal. Appl., 457 (2018), 672–693, arXiv: 1605.06785. doi: 10.1016/j.jmaa.2017.07.065. Google Scholar
[34]	M. Hinz, M. Röckner and A. Teplyaev, Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces, Stochastic Process. Appl., 123 (2013), 4373-4406. doi: 10.1016/j.spa.2013.06.009. Google Scholar
[35]	M. Hinz and A. Teplyaev, Dirac and magnetic Schrödinger operators on fractals, J. Funct. Anal., 265 (2013), 2830-2854. doi: 10.1016/j.jfa.2013.07.021. Google Scholar
[36]	M. Ionescu, L. Rogers and A. Teplyaev, Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169. doi: 10.1016/j.jfa.2012.05.021. Google Scholar
[37]	V. A. Kaimanovich, "Münchhausen trick" and amenability of self-similar groups, Internat. J. Algebra Comput., 15 (2005), 907-937. doi: 10.1142/S0218196705002694. Google Scholar
[38]	N. Kajino, Heat kernel asymptotics for the measurable Riemannian structure on the Sierpinski gasket, Potential Anal., 36 (2012), 67-115. doi: 10.1007/s11118-011-9221-5. Google Scholar
[39]	N. Kajino, Analysis and geometry of the measurable Riemannian structure on the Sierpiński gasket, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. I. Fractals in Pure Mathematics, vol. 600 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2013, 91–133. doi: 10.1090/conm/600/11932. Google Scholar
[40]	C. J. Kauffman, R. M. Kesler, A. G. Parshall, E. A. Stamey and B. A. Steinhurst, Quantum mechanics on Laakso spaces, J. Math. Phys., 53(2012), 042102, 18pp. doi: 10.1063/1.3702099. Google Scholar
[41]	D. J. Kelleher, B. A. Steinhurst and C. -M. M. Wong, From self-similar structures to selfsimilar groups, Internat. J. Algebra Comput., 22 (2012), 1250056, 16pp. doi: 10.1142/S0218196712500567. Google Scholar
[42]	M. Keller, D. Lenz and R. K. Wojciechowski, Volume growth, spectrum and stochastic completeness of infinite graphs, Math. Z., 274 (2013), 905-932. doi: 10.1007/s00209-012-1101-1. Google Scholar
[43]	J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755. doi: 10.2307/2154402. Google Scholar
[44]	J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. Google Scholar
[45]	J. Kigami, Local Nash inequality and inhomogeneity of heat kernels, Proc. London Math. Soc. (3), 89 (2004), 525-544. doi: 10.1112/S0024611504014807. Google Scholar
[46]	J. Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc., 199 (2009), ⅷ+94pp. doi: 10.1090/memo/0932. Google Scholar
[47]	J. Kigami, Quasisymmetric modification of metrics on self-similar sets, in Geometry and Analysis of Fractals, vol. 88 of Springer Proc. Math. Stat., Springer, Heidelberg, 2014,253–282. doi: 10.1007/978-3-662-43920-3_9. Google Scholar
[48]	O. Knill, The graph spectrum of barycentric refinements, arXiv: 1508.02027. Google Scholar
[49]	O. Knill, Universality for Barycentric subdivision, arXiv: 1509.06092. Google Scholar
[50]	S. Kusuoka and X. Y. Zhou, Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Related Fields, 93 (1992), 169-196. doi: 10.1007/BF01195228. Google Scholar
[51]	M. Lapidus and J. Sarhad, Dirac operators and geodesic metric on the harmonic Sierpinski gasket and other fractal sets, J. Noncommut. Geom., 8 (2014), 947-985. doi: 10.4171/JNCG/174. Google Scholar
[52]	P. Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2), 124 (1986), 1-21. doi: 10.2307/1971385. Google Scholar
[53]	P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201. doi: 10.1007/BF02399203. Google Scholar
[54]	T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc., 83 (1990), ⅳ+128pp. doi: 10.1090/memo/0420. Google Scholar
[55]	D. Lougee and B. Steinhurst, Bond percolation on a non-P. C. F. Sierpiński gasket, iterated barycentric subdivision of a triangle, and hexacarpet, Fractals, 24 (2016), 1650023, 12pp. doi: 10.1142/S0218348X16500237. Google Scholar
[56]	R. Lyons and Y. Peres, Probability on Trees and Networks, vol. 42 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, New York, 2016, Available at http://pages.iu.edu/~rdlyons/. doi: 10.1017/9781316672815. Google Scholar
[57]	I. McGillivray, Resistance in higher-dimensional Sierpiński carpets, Potential Anal., 16 (2002), 289-303. doi: 10.1023/A:1014035414658. Google Scholar
[58]	D. Molitor, N. Ott and R. Strichartz, Using Peano curves to construct Laplacians on fractals, Fractals, 23 (2015), 1550048, 29pp. doi: 10.1142/S0218348X15500486. Google Scholar
[59]	V. Nekrashevych, Self-similar Groups, vol. 117 of Mathematical Surveys and Monographs, Amer. Math. Soc., 2005. doi: 10.1090/surv/117. Google Scholar
[60]	V. Nekrashevych and A. Teplyaev, Groups and analysis on fractals, in Analysis on Graphs and Its Applications, vol. 77 of Proc. Sympos. Pure Math., Amer. Math. Soc., 2008, 143–180. doi: 10.1090/pspum/077/2459868. Google Scholar
[61]	L. Rogers and A. Teplyaev, Laplacians on the basilica Julia sets, Commun. Pure Appl. Anal., 9 (2010), 211-231. doi: 10.3934/cpaa.2010.9.211. Google Scholar
[62]	B. Steinhurst, Uniqueness of locally symmetric Brownian motion on Laakso spaces, Potential Anal., 38 (2013), 281-298. doi: 10.1007/s11118-012-9273-1. Google Scholar
[63]	R. S. Strichartz, Differential Equations on Fractals. A Tutorial, Princeton Univ. Press, 2006. Google Scholar
[64]	A. Telcs and V. Vespri, Resolvent metric and the heat kernel estimate for random walks, Stochastic Process. Appl., 124 (2014), 3965-3985. doi: 10.1016/j.spa.2014.07.012. Google Scholar
[65]	A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure, Canad. J. Math., 60 (2008), 457-480. doi: 10.4153/CJM-2008-022-3. Google Scholar
[66]	S. Volkov, Random geometric subdivisions, Random Structures Algorithms, 43 (2013), 115-130. doi: 10.1002/rsa.20454. Google Scholar

Figure 1. On the left: barycentric subdivision of a 2-simplex, the graphs $G_0^T$, $G_1^T$ and $G_2^T$. On the right: adjacency (dual) graph $G_2$, in bold, pictured together with the thin image of $G_2^T$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. On the left: the graph $G_4^T$ for barycentric subdivision of a 2-simplex. On the right: the adjacency (dual) graph $G_4$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. $A_{(2)}^{T/H}$ and $B_{(2)}^{T/H}$ on the hexagonal embedding of $G_2^{T/H}$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. $\tilde G^H_2$ and $\tilde G_2^T$ without the additional edges

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. The transformation from the flow $I^n$ (left) to the flow $H_{02}^n$ (right)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. The function $u$, $v$ and $w$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. Gluing from $\widehat{G_1}$ to $\widehat{G_2}$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. Short-circuited graphs $\widetilde G^T_1$ and $\widetilde G^T_2$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9. Left: Graph $\widetilde G_2^H$ with short circuits. Right: Non-p.c.f Sierpinski gasket

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Weronika Biedrzycka, Marta Tyran-Kamińska. Self-similar solutions of fragmentation equations revisited. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 13-27. doi: 10.3934/dcdsb.2018002
[2]	Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801
[3]	Rostislav Grigorchuk, Volodymyr Nekrashevych. Self-similar groups, operator algebras and Schur complement. Journal of Modern Dynamics, 2007, 1 (3) : 323-370. doi: 10.3934/jmd.2007.1.323
[4]	Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[5]	Anna Chiara Lai, Paola Loreti. Self-similar control systems and applications to zygodactyl bird's foot. Networks & Heterogeneous Media, 2015, 10 (2) : 401-419. doi: 10.3934/nhm.2015.10.401
[6]	Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036
[7]	D. G. Aronson. Self-similar focusing in porous media: An explicit calculation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1685-1691. doi: 10.3934/dcdsb.2012.17.1685
[8]	G. A. Braga, Frederico Furtado, Vincenzo Isaia. Renormalization group calculation of asymptotically self-similar dynamics. Conference Publications, 2005, 2005 (Special) : 131-141. doi: 10.3934/proc.2005.2005.131
[9]	Qiaolin He. Numerical simulation and self-similar analysis of singular solutions of Prandtl equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 101-116. doi: 10.3934/dcdsb.2010.13.101
[10]	Bendong Lou. Self-similar solutions in a sector for a quasilinear parabolic equation. Networks & Heterogeneous Media, 2012, 7 (4) : 857-879. doi: 10.3934/nhm.2012.7.857
[11]	F. Berezovskaya, G. Karev. Bifurcations of self-similar solutions of the Fokker-Plank equations. Conference Publications, 2005, 2005 (Special) : 91-99. doi: 10.3934/proc.2005.2005.91
[12]	Shota Sato, Eiji Yanagida. Singular backward self-similar solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 897-906. doi: 10.3934/dcdss.2011.4.897
[13]	Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[14]	Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[15]	L. Olsen. Rates of convergence towards the boundary of a self-similar set. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 799-811. doi: 10.3934/dcds.2007.19.799
[16]	Marek Fila, Michael Winkler, Eiji Yanagida. Convergence to self-similar solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 703-716. doi: 10.3934/dcds.2008.21.703
[17]	Hyungjin Huh. Self-similar solutions to nonlinear Dirac equations and an application to nonuniqueness. Evolution Equations & Control Theory, 2018, 7 (1) : 53-60. doi: 10.3934/eect.2018003
[18]	Kumiko Hattori, Noriaki Ogo, Takafumi Otsuka. A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 289-311. doi: 10.3934/dcdss.2017014
[19]	Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[20]	K. T. Joseph, Philippe G. LeFloch. Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits. Communications on Pure & Applied Analysis, 2002, 1 (1) : 51-76. doi: 10.3934/cpaa.2002.1.51

2019 Impact Factor: 1.233

Tools

Metrics

Tools

Article outline

Figures and Tables

2019 Impact Factor: 1.233

Tools

Figures and Tables

2019 Impact Factor: 1.233

American Institute of Mathematical Sciences

Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Dual graphs and modified Barlow-Bass resistance estimates for repeated barycentric subdivisions

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors