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March  2014, 13(2): 903-928. doi: 10.3934/cpaa.2014.13.903

Hodge-de Rham theory on fractal graphs and fractals

1. 

Reed College, Oregon, United States

2. 

Rice University, Texas, United States

3. 

Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853, United States

4. 

The Chinese University of Hong Kong

Received  November 2012 Revised  May 2013 Published  October 2013

We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and 3-dimensional Sierpinski gasket (SG$^3$), but the method is expected to be effective for many PCF fractals, and also infinitely ramified fractals such as the Sierpinski carpet (SC). Our approach is to construct k-forms and de Rham differential operators $d$ and $\delta$ for a sequence of graphs approximating the fractal, and then pass to the limit with suitable renormalization, in imitation of Kigami's approach on constructing Laplacians on functions. One of our results is that our Laplacian on 0-forms is equal to Kigami's Laplacian on functions. We give explicit construction of harmonic 1-forms for our examples. We also prove that the measures on line segments provided by 1-forms are not absolutely continuous with respect to Lebesgue measures.
Citation: S. Aaron, Z. Conn, Robert S. Strichartz, H. Yu. Hodge-de Rham theory on fractal graphs and fractals. Communications on Pure and Applied Analysis, 2014, 13 (2) : 903-928. doi: 10.3934/cpaa.2014.13.903
References:
[1]

J. Azzam, M. Hall and R. S. Strichartz, Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket, Transactions of the American Mathematical Society, 360 (2008), 2089-2130. doi: 10.1090/S0002-9947-07-04363-2.

[2]

M. T. Barlow, "Diffusions on Fractals," L.N.M. 1690, Springer-Verlag, New York, 1999, 1-112. doi: 10.1007/BFb0092537.

[3]

J. Bello, Y. Li and R. S. Strichartz, Hodge-de Rham theory of k-forms on carpet type fractals,, in preparation., (). 

[4]

F. Cipriani, Diriclet forms on noncommutative spaces, L.N.M. "Quantum Potential Theory," 1954, U. Franz-M Schurmann eds. Springer-Verlag, New York, 2008, 161-172. doi: 10.1007/978-3-540-69365-9_5.

[5]

F. Cipriani, D Guido, T. Isola and J. Sauvageot, Spectral triples on the Sierpinski gasket, in AMS Meeting "Analysis, Probability and Mathematical Physics on Fractals," Cornell U., 2011.

[6]

F. Cipriani, D Guido, T, Isola and J. Sauvageot, Differential 1-forms, their integral and potential theory on the Sierpinski gasket,, arXiv:1105.1995., (). 

[7]

F. Cipriani and J. Sauvageot, Derivations as square roots of Dirichlet forms, J. Funct. Ana., 201 (2003), 78-120. doi: 10.1016/S0022-1236(03)00085-5.

[8]

Colin de Verdière, "Spectres de graphes, Cours spécialisés," vol. 4, Paris: Société Mathématique de France, 1998.

[9]

M. Cucuringu and R. S. Strichartz, Self-similar energy forms on the Sierpinski gasket with twists, Potential Anal., 27 (2007), 45-60. doi: 10.1007/s11118-007-9047-3.

[10]

K. Dalrymple, R. S. Strichartz and J. P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Funct. Ana. and App., 5 (1999). doi: 10.1007/BF01261610.

[11]

D Guido and T. Isola, Singular traces on semi-finite von Neumann algebras, J. Funct. Ana., 134 (1995), 451-485. doi: 10.1006/jfan.1995.1153.

[12]

D. Guido and T. Isola, Dimensions and singular traces for spectral triples, with applications to fractals, J. Funct. Anal., 203 (2003), 362-400. doi: 10.1016/S0022-1236(03)00230-1.

[13]

D. Guido and T. Isola, Dimensions and singular traces for spectral triples for fratcals in $\mathbbR^N$, Advances in Operator Algebras and Mathematical Physics; Proceedings of the Conference held in Sinaia, Romania, June 2003, F. Boca, O. Bratteli, R. Longo, H. Siedentop Eds., Theta Series in Advanced Mathematics, Bucharest 2005.

[14]

M. Hinz, Limit chains on the Sierpinski gasket,, Indiana U. Math. J., ().  doi: 10.1512/iumj.2011.60.4404.

[15]

M. Ionescu, L. G. Rogers and A. Teplyaev, Derivations and Dirichlet forms on fractals,, arXiv:1106.1450., ().  doi: 10.1016/j.jfa.2012.05.021.

[16]

J. Kigami, "Anaysis on Fractals," Cambridge Tracts in Mathematics, 143, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943.

[17]

J. Owen and R. S. Strichartz, Boundary value problems for harmonic functions on a domain in the Sierpinski gasket, Indiana U. Math. J., 61(2012), 319-335. doi: 10.1512/iumj.2012.61.4539.

[18]

R. Peirone, Existence of eigenforms on fractals with three vertices, Proc. Royal Soc. Edinburgh, 137A (2007), 1073-1080. doi: 10.1017/S0308210505001137.

[19]

R. Peirone, Existence of eigenforms on nicely separated fractals, Proc. of Symposia in Pure Math., Amer. Math. Soc., vol. 77, 231-241, 2008.

[20]

C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. Sci. École Norm. Sup., 30 (1997), 605-673. doi: 10.1016/S0012-9593(97)89934-X.

[21]

R. S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127. doi: 10.1006/jfan.2000.3580.

[22]

R. S. Strichartz, Harmonic mappings of the Sierpinski gasket to the circle, Proceedings of the American Mathematical Society, 130 (2001), 805-817. doi: 10.1090/S0002-9939-01-06243-8.

[23]

R. S. Strichartz, "Differential Equations on Fractals: A Tutorial," Princeton Univ. Press, Princeton 2006.

[24]

A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure, Canadian Journal of Math., 60 (2008), 457-480. doi: 10.4153/CJM-2008-022-3.

show all references

References:
[1]

J. Azzam, M. Hall and R. S. Strichartz, Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket, Transactions of the American Mathematical Society, 360 (2008), 2089-2130. doi: 10.1090/S0002-9947-07-04363-2.

[2]

M. T. Barlow, "Diffusions on Fractals," L.N.M. 1690, Springer-Verlag, New York, 1999, 1-112. doi: 10.1007/BFb0092537.

[3]

J. Bello, Y. Li and R. S. Strichartz, Hodge-de Rham theory of k-forms on carpet type fractals,, in preparation., (). 

[4]

F. Cipriani, Diriclet forms on noncommutative spaces, L.N.M. "Quantum Potential Theory," 1954, U. Franz-M Schurmann eds. Springer-Verlag, New York, 2008, 161-172. doi: 10.1007/978-3-540-69365-9_5.

[5]

F. Cipriani, D Guido, T. Isola and J. Sauvageot, Spectral triples on the Sierpinski gasket, in AMS Meeting "Analysis, Probability and Mathematical Physics on Fractals," Cornell U., 2011.

[6]

F. Cipriani, D Guido, T, Isola and J. Sauvageot, Differential 1-forms, their integral and potential theory on the Sierpinski gasket,, arXiv:1105.1995., (). 

[7]

F. Cipriani and J. Sauvageot, Derivations as square roots of Dirichlet forms, J. Funct. Ana., 201 (2003), 78-120. doi: 10.1016/S0022-1236(03)00085-5.

[8]

Colin de Verdière, "Spectres de graphes, Cours spécialisés," vol. 4, Paris: Société Mathématique de France, 1998.

[9]

M. Cucuringu and R. S. Strichartz, Self-similar energy forms on the Sierpinski gasket with twists, Potential Anal., 27 (2007), 45-60. doi: 10.1007/s11118-007-9047-3.

[10]

K. Dalrymple, R. S. Strichartz and J. P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Funct. Ana. and App., 5 (1999). doi: 10.1007/BF01261610.

[11]

D Guido and T. Isola, Singular traces on semi-finite von Neumann algebras, J. Funct. Ana., 134 (1995), 451-485. doi: 10.1006/jfan.1995.1153.

[12]

D. Guido and T. Isola, Dimensions and singular traces for spectral triples, with applications to fractals, J. Funct. Anal., 203 (2003), 362-400. doi: 10.1016/S0022-1236(03)00230-1.

[13]

D. Guido and T. Isola, Dimensions and singular traces for spectral triples for fratcals in $\mathbbR^N$, Advances in Operator Algebras and Mathematical Physics; Proceedings of the Conference held in Sinaia, Romania, June 2003, F. Boca, O. Bratteli, R. Longo, H. Siedentop Eds., Theta Series in Advanced Mathematics, Bucharest 2005.

[14]

M. Hinz, Limit chains on the Sierpinski gasket,, Indiana U. Math. J., ().  doi: 10.1512/iumj.2011.60.4404.

[15]

M. Ionescu, L. G. Rogers and A. Teplyaev, Derivations and Dirichlet forms on fractals,, arXiv:1106.1450., ().  doi: 10.1016/j.jfa.2012.05.021.

[16]

J. Kigami, "Anaysis on Fractals," Cambridge Tracts in Mathematics, 143, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943.

[17]

J. Owen and R. S. Strichartz, Boundary value problems for harmonic functions on a domain in the Sierpinski gasket, Indiana U. Math. J., 61(2012), 319-335. doi: 10.1512/iumj.2012.61.4539.

[18]

R. Peirone, Existence of eigenforms on fractals with three vertices, Proc. Royal Soc. Edinburgh, 137A (2007), 1073-1080. doi: 10.1017/S0308210505001137.

[19]

R. Peirone, Existence of eigenforms on nicely separated fractals, Proc. of Symposia in Pure Math., Amer. Math. Soc., vol. 77, 231-241, 2008.

[20]

C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. Sci. École Norm. Sup., 30 (1997), 605-673. doi: 10.1016/S0012-9593(97)89934-X.

[21]

R. S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127. doi: 10.1006/jfan.2000.3580.

[22]

R. S. Strichartz, Harmonic mappings of the Sierpinski gasket to the circle, Proceedings of the American Mathematical Society, 130 (2001), 805-817. doi: 10.1090/S0002-9939-01-06243-8.

[23]

R. S. Strichartz, "Differential Equations on Fractals: A Tutorial," Princeton Univ. Press, Princeton 2006.

[24]

A. Teplyaev, Harmonic coordinates on fractals with finitely ramified cell structure, Canadian Journal of Math., 60 (2008), 457-480. doi: 10.4153/CJM-2008-022-3.

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