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Convergence of Energy Forms on Sierpinski Gaskets with Added Rotated Triangle

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Abstract

We study the convergence of resistance metrics and resistance forms on a converging sequence of spaces. As an application, we study the existence and uniqueness of self-similar Dirichlet forms on Sierpinski gaskets with added rotated triangle. The fractals depend on a parameter in a continuous way. When the parameter is irrational, the fractal is not post critically finite (p.c.f.), and there are infinitely many ways that two cells intersect. In this case, we define the Dirichlet form as a limit in some Γ-convergence sense of the Dirichlet forms on p.c.f. fractals that approximate it.

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References

  1. Alexander, S., Orbach, R.: Density of states on fractals: “fractons”. J. Physique (Paris) Lett. 43, 625–631 (1982)

    Google Scholar 

  2. Alonso Ruiz, P., Freiberg, U., Kigami, J.: Completely symmetric resistance forms on the stretched Sierpiński gasket. J. Fractal Geom. 5(3), 227–277 (2018)

    MathSciNet  Google Scholar 

  3. Barlow, M.T.: Diffusions on Fractals. Lectures on Probability Theory and Statistics (Saint-Flour, 1995), 1–121, Lecture Notes in Math. 1690. Springer, Berlin (1998)

    Google Scholar 

  4. Barlow, M.T., Bass, R.F.: The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. Henri Poincaré 25(3), 225–257 (1989)

    MathSciNet  Google Scholar 

  5. Barlow, M.T., Bass, R.F.: Transition densities for Brownian motion on the Sierpinski carpet. Probab. Theory Related Fields 91, 307–330 (1992)

    MathSciNet  Google Scholar 

  6. Barlow, M.T., Bass, R.F.: Brownian motion and harmonic analysis on Sierpinski carpets. Canad. J. Math. 51(4), 673–744 (1999)

    MathSciNet  Google Scholar 

  7. Barlow, M.T., Bass, R.F., Kumagai, T., Teplyaev, A.: Uniqueness of Brownian motion on Sierpinski carpets. J. Eur. Math. Soc. 12(3), 655–701 (2010)

    MathSciNet  Google Scholar 

  8. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence (2001)

    Google Scholar 

  9. Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpiński gasket. Probab. Theory Related Fields 79(4), 543–623 (1988)

    MathSciNet  Google Scholar 

  10. Croydon, D.A.: Scaling limits of stochastic processes associated with resistance forms. Ann. Inst. Henri Poincaré, Probab. Stat. 54(4), 1939–1968 (2018)

    MathSciNet  Google Scholar 

  11. Croydon, D.A.: The Random Conductance Model with Heavy Tails on Nested Fractal Graphs. Fractal Geometry and Stochastics VI, 239–254, Progr. Probab. 76. Birkhäuser/Springer, Cham (2021)

    Google Scholar 

  12. Croydon, D.A., Hambly, B., Kumagai, T.: Time-changes of stochastic processes associated with resistance forms. Electron. J. Probab. 22(82), 41 (2017)

    MathSciNet  Google Scholar 

  13. Croydon, D.A., Hambly, B., Kumagai, T.: Heat kernel estimates for FIN processes associated with resistance forms. Stochastic Process. Appl. 129 (9), 2991–3017 (2019)

    MathSciNet  Google Scholar 

  14. Cao, S., Qiu, H.: Resistance forms on self-similar sets with finite ramification of finite type. Potential Anal. 54(4), 581–606 (2021)

    MathSciNet  Google Scholar 

  15. Cao, S., Qiu, H.: Brownian motion on the golden ratio Sierpinski gasket. arXiv:2010.05181 (2020)

  16. Cao, S., Hassler, M., Qiu, H., Sandine, E., Strichartz, R.S.: Existence and uniqueness of diffusions on the Julia sets of Misiurewicz-Sierpinski maps. Adv. Math. 389, Paper No. 107922, 41pp (2021)

  17. Dal Maso, G.: An Introduction to Γ-Convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston (1993)

    Google Scholar 

  18. Elia, M., Peirone, R.: Eigenforms on fractals with connected interior and three vertices. Fractals 26(4), 1850082 (2018)

    MathSciNet  Google Scholar 

  19. Flock, T.C., Strichartz, R.C.: Laplacians on a family of quadratic Julia sets I. Trans. Amer. Math. Soc. 364(8), 3915–3965 (2012)

    MathSciNet  Google Scholar 

  20. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Second Revised and Extended Edition. De Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin (2011)

    Google Scholar 

  21. Gigli, N., Mondino, A., Savaré, G.: Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows. Proc. Lond. Math. Soc. (3) 111(5), 1071–1129 (2015)

    MathSciNet  Google Scholar 

  22. Goldstein, S.: Random Walks and Diffusions on Fractals. Percolation Theory and Ergodic Theory of Infinite Particle Systems (Minneapolis, Minn., 1984–1985), 121–129, IMA Vol. Math. Appl., 8. Springer, New York (1987)

    Google Scholar 

  23. Hambly, B.M., Kumagai, T.: Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. London Math. Soc. (3) 78(2), 431–458 (1999)

    MathSciNet  Google Scholar 

  24. Hambly, B.M., Nyberg, S.O.G.: Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem. Proc. Edinb. Math. Soc. (2) 46(1), 1–34 (2003)

    MathSciNet  Google Scholar 

  25. Havlin, S., Ben-Avarham, D.: Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987)

    Google Scholar 

  26. Hino, M.: On singularity of energy measures on self-similar sets. Probab. Theory Related Fields 132(2), 265–290 (2005)

    MathSciNet  Google Scholar 

  27. Hambly, B.M., Metz, V., Teplyaev, A.: Self-similar energies on post-critically finite self-similar fractals. J. London Math. Soc. 74, 93–112 (2006)

    MathSciNet  Google Scholar 

  28. Kigami, J.: A harmonic calculus on the Sierpinski spaces. Japan J. Appl. Math. 6(2), 259–290 (1989)

    MathSciNet  Google Scholar 

  29. Kigami, J.: A harmonic calculus on p.c.f. self-similar sets. Trans. Amer. Math. Soc. 335(2), 721–755 (1993)

    MathSciNet  Google Scholar 

  30. Kigami, J.: Analysis on Fractals. Cambridge Tracts in Mathematics 143. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  31. Kigami, J.: Resistance forms, quasisymmetric maps and heat kernel estimates. Mem. Amer. Math. Soc. 216(1015), vi+ 132 (2012)

    MathSciNet  Google Scholar 

  32. Kumagai, T.: Regularity, closedness and spectral dimensions of the Dirichlet forms on P.C.F. self-similar sets. J. Math. Kyoto Univ. 33(3), 765–786 (1993)

    MathSciNet  Google Scholar 

  33. Kusuoka, S.: A diffusion process on a fractal. In: Ito, K., Ikeda, N. (eds.) Probabilistic Methods in Mathematical Physics, Pro. Taniguchi Intern. Symp. (Katata/Kyoto, 1985), pp. 251–274. Academic Press, Boston (1987)

  34. Kusuoka, S., Zhou, X.Y.: Dirichlet forms on fractals: Poincaré constant and resistance. Probab. Theory Related Fields 93(2), 169–196 (1992)

    MathSciNet  Google Scholar 

  35. Kuwae, K., Shioya, T.: Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. Comm. Anal. Geom. 11 (4), 599–673 (2003)

    MathSciNet  Google Scholar 

  36. Lindstrøm, T.: Brownian motion on nested fractals. Mem. Amer. Math. Soc. 83(420), iv+ 128 (1990)

    MathSciNet  Google Scholar 

  37. Metz, V.: Hilbert’s projective metric on cones of Dirichlet forms. J. Funct. Anal. 127(2), 438–455 (1995)

    MathSciNet  Google Scholar 

  38. Metz, V.: Renormalization contracts on nested fractals. J. Reine Angew. Math. 480, 161–175 (1996)

    MathSciNet  Google Scholar 

  39. Metz, V.: Nonlinear Perron-Frobenius theory in finite dimensions. Nonlinear Anal. 62(2), 225–244 (2005)

    MathSciNet  Google Scholar 

  40. Metz, V.: The short-cut test. J. Funct. Anal. 220(1), 118–156 (2005)

    MathSciNet  Google Scholar 

  41. Metz, V., Grabner, P.: An interface problem for a Sierpinski and a Vicsek fractal. Math. Nachr. 280(13-14), 1577–1594 (2007)

    MathSciNet  Google Scholar 

  42. Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2), 368–421 (1994)

    MathSciNet  Google Scholar 

  43. Peirone, R.: Convergence and uniqueness problems for Dirichlet forms on fractals. Boll. Unione Mat. Ital. Sez. B (8) 3, 431–460 (2000)

    MathSciNet  Google Scholar 

  44. Peirone, R.: Uniqueness of eigenforms on fractals—II. Math. Nachr. 288(11-12), 1431–1447 (2015)

    MathSciNet  Google Scholar 

  45. Peirone, R.: Scaling distances on finitely ramified fractals. Kyoto J. Math. 57(3), 475–504 (2017)

    MathSciNet  Google Scholar 

  46. Peirone, R.: A p.c.f. self-similar set with no self-similar energy. J. Fractal Geom. 6(4), 393–404 (2019)

    MathSciNet  Google Scholar 

  47. Peirone, R.: Fixed points of anti-attracting maps and eigenforms on fractals. Math. Nachr. 294(8), 1578–1594 (2021)

    MathSciNet  Google Scholar 

  48. Rogers, L.G., Teplyaev, A.: Laplacians on the Basilica Julia sets. Comm. Pure Appl. Anal. 9(1), 211–231 (2010)

    MathSciNet  Google Scholar 

  49. Rammal, R., Toulouse, G.: Random walks on fractal structures and percolation clusters. J. Physique Lettres 44, 13–22 (1983)

    Google Scholar 

  50. Sabot, C.: Existence and uniqueness of diffusions on finitely ramified self-similar fractals (English, French summary). Ann. Sci. École Norm. Sup. 30(5), 605–673 (1997)

    MathSciNet  Google Scholar 

  51. Strichartz, R.S.: Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton (2006)

    Google Scholar 

  52. Teplyaev, A.: Harmonic coordinates on fractals with finitely ramified cell structure. Canad. J. Math. 60(2), 457–480 (2008)

    MathSciNet  Google Scholar 

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Acknowledgments

The author is grateful to Professor Robert S. Strichartz for his continued support and encouragement for me to work on this problem. He also wants to thank the anonymous referees for their suggestions on improving the writing of the paper.

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  1. Department of Mathematics, Cornell University, Ithaca, 14853, USA

    Shiping Cao

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Cao, S. Convergence of Energy Forms on Sierpinski Gaskets with Added Rotated Triangle. Potential Anal 59, 1793–1825 (2023). https://doi.org/10.1007/s11118-022-10034-9

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