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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 & 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.  doi: 10.1090/S0002-9947-07-04363-2.  Google Scholar

[2]

M. T. Barlow, "Diffusions on Fractals,", L.N.M. 1690, (1690), 1.  doi: 10.1007/BFb0092537.  Google Scholar

[3]

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

[4]

F. Cipriani, Diriclet forms on noncommutative spaces,, L.N.M., (1954), 161.  doi: 10.1007/978-3-540-69365-9_5.  Google Scholar

[5]

F. Cipriani, D Guido, T. Isola and J. Sauvageot, Spectral triples on the Sierpinski gasket,, in AMS Meeting, (2011).   Google Scholar

[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., ().   Google Scholar

[7]

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

[8]

Colin de Verdière, "Spectres de graphes, Cours spécialisés,", vol. 4, (1998).   Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[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, (2003).   Google Scholar

[14]

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

[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.  Google Scholar

[16]

J. Kigami, "Anaysis on Fractals,", Cambridge Tracts in Mathematics, (2001).  doi: 10.1017/CBO9780511470943.  Google Scholar

[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.  doi: 10.1512/iumj.2012.61.4539.  Google Scholar

[18]

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

[19]

R. Peirone, Existence of eigenforms on nicely separated fractals,, Proc. of Symposia in Pure Math., (2008), 231.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

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.  doi: 10.1090/S0002-9947-07-04363-2.  Google Scholar

[2]

M. T. Barlow, "Diffusions on Fractals,", L.N.M. 1690, (1690), 1.  doi: 10.1007/BFb0092537.  Google Scholar

[3]

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

[4]

F. Cipriani, Diriclet forms on noncommutative spaces,, L.N.M., (1954), 161.  doi: 10.1007/978-3-540-69365-9_5.  Google Scholar

[5]

F. Cipriani, D Guido, T. Isola and J. Sauvageot, Spectral triples on the Sierpinski gasket,, in AMS Meeting, (2011).   Google Scholar

[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., ().   Google Scholar

[7]

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

[8]

Colin de Verdière, "Spectres de graphes, Cours spécialisés,", vol. 4, (1998).   Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[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, (2003).   Google Scholar

[14]

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

[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.  Google Scholar

[16]

J. Kigami, "Anaysis on Fractals,", Cambridge Tracts in Mathematics, (2001).  doi: 10.1017/CBO9780511470943.  Google Scholar

[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.  doi: 10.1512/iumj.2012.61.4539.  Google Scholar

[18]

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

[19]

R. Peirone, Existence of eigenforms on nicely separated fractals,, Proc. of Symposia in Pure Math., (2008), 231.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

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