American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Quasilinear retarded differential equations with functional dependence on piecewise constant argument
  • CPAA Home
  • This Issue
  • Next Article
    Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type
March  2014, 13(2): 903-928. doi: 10.3934/cpaa.2014.13.903

Hodge-de Rham theory on fractal graphs and fractals

S. Aaron 1, , Z. Conn 2, , Robert S. Strichartz 3, and  H. Yu 4,

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.
Keywords: Analysis on fractals, Sierpinski gasket, k-forms, harmonic 1-forms, Hodge-de Rham theory, fractal graphs..
Mathematics Subject Classification: 28A8.
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), 2089-2130. doi: 10.1090/S0002-9947-07-04363-2.  Google Scholar

[2]

M. T. Barlow, "Diffusions on Fractals," L.N.M. 1690, Springer-Verlag, New York, 1999, 1-112. 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. "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.  Google Scholar

[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. 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), 78-120. doi: 10.1016/S0022-1236(03)00085-5.  Google Scholar

[8]

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

[9]

M. Cucuringu and R. S. Strichartz, Self-similar energy forms on the Sierpinski gasket with twists, Potential Anal., 27 (2007), 45-60. 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), 451-485. 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), 362-400. 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, Romania, June 2003, F. Boca, O. Bratteli, R. Longo, H. Siedentop Eds., Theta Series in Advanced Mathematics, Bucharest 2005.  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, 143, Cambridge University Press, Cambridge, 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), 319-335. 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), 1073-1080. doi: 10.1017/S0308210505001137.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

R. S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127. 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), 805-817. doi: 10.1090/S0002-9939-01-06243-8.  Google Scholar

[23]

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

[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.  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), 2089-2130. doi: 10.1090/S0002-9947-07-04363-2.  Google Scholar

[2]

M. T. Barlow, "Diffusions on Fractals," L.N.M. 1690, Springer-Verlag, New York, 1999, 1-112. 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. "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.  Google Scholar

[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. 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), 78-120. doi: 10.1016/S0022-1236(03)00085-5.  Google Scholar

[8]

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

[9]

M. Cucuringu and R. S. Strichartz, Self-similar energy forms on the Sierpinski gasket with twists, Potential Anal., 27 (2007), 45-60. 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), 451-485. 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), 362-400. 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, Romania, June 2003, F. Boca, O. Bratteli, R. Longo, H. Siedentop Eds., Theta Series in Advanced Mathematics, Bucharest 2005.  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, 143, Cambridge University Press, Cambridge, 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), 319-335. 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), 1073-1080. doi: 10.1017/S0308210505001137.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

R. S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127. 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), 805-817. doi: 10.1090/S0002-9939-01-06243-8.  Google Scholar

[23]

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

[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.  Google Scholar

[1]

Jessica Hyde, Daniel Kelleher, Jesse Moeller, Luke Rogers, Luis Seda. Magnetic Laplacians of locally exact forms on the Sierpinski Gasket. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2299-2319. doi: 10.3934/cpaa.2017113
[2]

Luigi Fontana, Steven G. Krantz and Marco M. Peloso. Hodge theory in the Sobolev topology for the de Rham complex on a smoothly bounded domain in Euclidean space. Electronic Research Announcements, 1995, 1: 103-107.

[3]

Jiahui Chen, Rundong Zhao, Yiying Tong, Guo-Wei Wei. Evolutionary de Rham-Hodge method. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3785-3821. doi: 10.3934/dcdsb.2020257
[4]

Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159
[5]

Yuri B. Suris. Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms. Journal of Geometric Mechanics, 2013, 5 (3) : 365-379. doi: 10.3934/jgm.2013.5.365
[6]

Huai-Dong Cao and Jian Zhou. On quantum de Rham cohomology theory. Electronic Research Announcements, 1999, 5: 24-34.

[7]

Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42.

[8]

R.D.S. Oliveira, F. Tari. On pairs of differential $1$-forms in the plane. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 519-536. doi: 10.3934/dcds.2000.6.519
[9]

Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691
[10]

Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.

[11]

Shiping Cao, Shuangping Li, Robert S. Strichartz, Prem Talwai. A trace theorem for Sobolev spaces on the Sierpinski gasket. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3901-3916. doi: 10.3934/cpaa.2020159
[12]

Saugata Bandyopadhyay, Bernard Dacorogna, Olivier Kneuss. The Pullback equation for degenerate forms. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 657-691. doi: 10.3934/dcds.2010.27.657
[13]

Yuval Z. Flicker. Automorphic forms on PGSp(2). Electronic Research Announcements, 2004, 10: 39-50.

[14]

Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623
[15]

P. Alonso Ruiz, Y. Chen, H. Gu, R. S. Strichartz, Z. Zhou. Analysis on hybrid fractals. Communications on Pure & Applied Analysis, 2020, 19 (1) : 47-84. doi: 10.3934/cpaa.2020004
[16]

Shiping Cao, Hua Qiu. Boundary value problems for harmonic functions on domains in Sierpinski gaskets. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1147-1179. doi: 10.3934/cpaa.2020054
[17]

Olivier Brahic. Infinitesimal gauge symmetries of closed forms. Journal of Geometric Mechanics, 2011, 3 (3) : 277-312. doi: 10.3934/jgm.2011.3.277
[18]

Anke D. Pohl. A dynamical approach to Maass cusp forms. Journal of Modern Dynamics, 2012, 6 (4) : 563-596. doi: 10.3934/jmd.2012.6.563
[19]

Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65
[20]

Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences