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Spectral asymptotics for \(V\)-variable Sierpinski gaskets. (English. French summary) Zbl 1387.35434

Summary: The family of \(V\)-variable fractals provides a means of interpolating between two families of random fractals previously considered in the literature; scale irregular fractals (\(V=1\)) and random recursive fractals (\(V=\infty\)). We consider a class of \(V\)-variable affine nested fractals based on the Sierpinski gasket with a general class of measures. We calculate the spectral exponent for a general measure and find the spectral dimension for these fractals. We show that the spectral properties and on-diagonal heat kernel estimates for \(V\)-variable fractals are closer to those of scale irregular fractals, in that it is the fluctuations in scale that determine their behaviour but that there are also effects of the spatial variability.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
28A80 Fractals
31C25 Dirichlet forms
35K08 Heat kernel
60J60 Diffusion processes
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[1] M. T. Barlow. Diffusions on fractals. InLectures in Probability Theory and Statistics: Ecole D’été de Probabilités de Saint-Flour XXV. Lect. Notes Math.1690. Springer, New York, 1998. · Zbl 0894.00045
[2] M. T. Barlow and B. M. Hambly. Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets.Ann. Inst. Henri Poincaré Probab. Stat.33(1997) 531-557. · Zbl 0903.60072 · doi:10.1016/S0246-0203(97)80104-5
[3] M. T. Barlow, A. A. Járai, T. Kumagai and G. Slade. Random walk on the incipient infinite cluster for oriented percolation in high dimensions.Comm. Math. Phys.278(2008) 385-431. · Zbl 1144.82030 · doi:10.1007/s00220-007-0410-4
[4] M. T. Barlow and T. Kumagai. Transition density asymptotics for some diffusion processes with multi-fractal structures.Electron. J. Probab.(2001).DOI:10.1214/EJP.v6-82. · Zbl 0974.60061
[5] M. Barnsley, J. E. Hutchinson and Ö. Stenflo. A fractal valued random iteration algorithm and fractal hierarchy.Fractals218(2005) 111-146. · Zbl 1304.28004 · doi:10.1142/S0218348X05002799
[6] M. Barnsley, J. E. Hutchinson and Ö. Stenflo. \(V\)-variable fractals: Fractals with partial self similarity.Adv. Math.18(2008) 2051-2088. · Zbl 1169.28006 · doi:10.1016/j.aim.2008.04.011
[7] M. Barnsley, J. E. Hutchinson and Ö. Stenflo. \(V\)-variable fractals: Dimension results.Forum Math.24(2012) 445-470. · Zbl 1244.28008
[8] D. A. Croydon. Heat kernel fluctuations for a resistance form with non-uniform volume growth.Proc. Lond. Math. Soc. (3)94(2007) 672-694. · Zbl 1116.58025
[9] D. Croydon and B. M. Hambly. Self-similarity and spectral asymptotics for the continuum random tree.Stochastic Process. Appl.118(2008) 730-754. · Zbl 1143.60012 · doi:10.1016/j.spa.2007.06.005
[10] S. Drenning and R. S. Strichartz. Spectral decimation on Hambly’s homogeneous hierarchical gaskets.Illinois J. Math.53(2009) 915-937. · Zbl 1211.28005
[11] K. J. Falconer. Random fractals.Math. Proc. Cambridge Philos. Soc.100(1986) 559-582. · Zbl 0623.60020 · doi:10.1017/S0305004100066299
[12] K. J. Falconer.Fractal Geometry. Mathematical Foundations and Applications, 2nd edition. Wiley, Hoboken, NJ, 2003. · Zbl 1060.28005
[13] P. J. Fitzsimmons, B. M. Hambly and T. Kumagai. Transition density estimates for Brownian motion on afline nested fractals.Comm. Math. Phys.165(1994) 595-620. · Zbl 0853.60062 · doi:10.1007/BF02099425
[14] M. Fukushima. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals. InIdeas and Methods in Mathematical Analysis, Stochastics, and Applications151-161.Oslo,1988. Cambridge Univ. Press, Cambridge, 1992.
[15] S. Graf. Statistically self-similar fractals.Probab. Theory Related Fields74(1987) 357-392. · Zbl 0591.60005 · doi:10.1007/BF00699096
[16] B. M. Hambly. Brownian motion on a homogeneous random fractal.Probab. Theory Related Fields94(1992) 1-38. · Zbl 0767.60075 · doi:10.1007/BF01222507
[17] B. M. Hambly. Brownian motion on a random recursive Sierpinski gasket.Ann. Probab.25(1997) 1059-1102. · Zbl 0895.60081 · doi:10.1214/aop/1024404506
[18] B. M. Hambly. Heat kernels and spectral asymptotics for some random Sierpinski gaskets. InFractal Geometry and Stochastics, II239-267.Greifswald/Koserow,1998. Progr. Probab.46. Birkhuser, Basel, 2000. · Zbl 0947.60086
[19] B. M. Hambly. On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets.Probab. Theory Related Fields117(2000) 221-247. · Zbl 0954.35121 · doi:10.1007/s004400050005
[20] B. M. Hambly, J. Kigami and T. Kumagai. Multifractal formalisms for the local spectral and walk dimensions.Math. Proc. Cambridge Philos. Soc.132(2002) 555-571. · Zbl 1002.60064
[21] B. M. Hambly and T. Kumagai. Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gaskets.Stochastic Process. Appl.92(2001) 61-85. · Zbl 1047.60084 · doi:10.1016/S0304-4149(00)00073-9
[22] B. M. Hambly and T. Kumagai. Diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice.Comm. Math. Phys.295(2010) 29-69. · Zbl 1191.82024 · doi:10.1007/s00220-009-0981-3
[23] B. M. Hambly, T. Kumagai, S. Kusuoka and X. Y. Zhou. Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets.J. Math. Soc. Japan52(2000) 373-408. · Zbl 0962.60078 · doi:10.2969/jmsj/05220373
[24] J. E. Hutchinson. Fractals and self-similarity.Indiana Univ. Math. J.30(1981) 713-747. · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[25] J. E. Hutchinson and L. Rüschendorf. Random fractals and probability metrics.Adv. in Appl. Probab.32(2000) 925-947. · Zbl 0984.60062 · doi:10.1017/S0001867800010375
[26] J. Kigami. Effective resistances for harmonic structures on p.c.f. self-similar sets.Math. Proc. Cambridge Philos. Soc.115(1994) 291-303. · Zbl 0803.60074 · doi:10.1017/S0305004100072091
[27] J. Kigami. Hausdorff dimensions of self-similar sets and shortest path metrics.J. Math. Soc. Japan47(1995) 381-404. · Zbl 0851.28002 · doi:10.2969/jmsj/04730381
[28] J. Kigami. Harmonic calculus on limits of networks and its application to dendrites.J. Funct. Anal.128(1995) 48-96. · Zbl 0820.60060 · doi:10.1006/jfan.1995.1023
[29] J. Kigami.Analysis on Fractals. Cambridge Univ. Press, Cambridge, 2001. · Zbl 0998.28004
[30] J. Kigami. Harmonic analysis for resistance forms.J. Funct. Anal.204(2003) 399-444. · Zbl 1039.31014 · doi:10.1016/S0022-1236(02)00149-0
[31] J. Kigami. Local Nash inequality and inhomogeneity of heat kernels.Proc. Lond. Math. Soc. (3)89(2004) 525-544. · Zbl 1060.60076 · doi:10.1112/S0024611504014807
[32] J. Kigami. Resistance forms, quasisymmetric maps and heat kernel estimates.Mem. Amer. Math. Soc.216(2012) 1015. · Zbl 1246.60099
[33] J. Kigami and M. L. Lapidus. Weyl’s problem for the spectral distribution of the Laplacian on P.C.F. self-similar fractals.Comm. Math. Phys.158(1993) 93-125. · Zbl 0806.35130 · doi:10.1007/BF02097233
[34] G. Kozma and A. Nachmias. The Alexander-Orbach conjecture holds in high dimensions.Invent. Math.178(2009) 635-654. · Zbl 1180.82094 · doi:10.1007/s00222-009-0208-4
[35] P. D. Lax.Functional Analysis. Wiley, New York, 2002. · Zbl 1009.47001
[36] R. D. Mauldin and S. C. Williams. Random recursive constructions: Asymptotic geometric and topological properties.Trans. Amer. Math. Soc.295(1986) 325-346. · Zbl 0625.54047 · doi:10.1090/S0002-9947-1986-0831202-5
[37] R. Scealy. \(V\)-variable fractals and interpolation. Ph.D. thesis, Australian National University, 2008.
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