메뉴 건너뛰기




Volumn 359, Issue 9, 2007, Pages 4339-4358

Spectral zeta functions of fractals and the complex dynamics of polynomials

Author keywords

Fractal; Fractal string; Laplacian; Rational complex dynamics; Spectral zeta function

Indexed keywords


EID: 37249077536     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-07-04150-5     Document Type: Article
Times cited : (51)

References (41)
  • 2
    • 0002163683 scopus 로고    scopus 로고
    • Diffusions on fractals
    • Saint-Flour, 1995 Lecture Notes in Math., 1690, Springer, Berlin MR1668115 (2000a:60148)
    • M.T. Barlow, Diffusions on fractals. Lectures on Probability Theory and Statistics (Saint-Flour, 1995), 1-121, Lecture Notes in Math., 1690, Springer, Berlin, 1998. MR1668115 (2000a:60148)
    • (1998) Lectures on Probability Theory and Statistics , pp. 1-121
    • Barlow, M.T.1
  • 3
    • 0001352912 scopus 로고    scopus 로고
    • Localized eigenfunctions of the Laplacian on p. c. f. self-similar sets
    • MR1489140 (99b:35162)
    • M.T. Barlow and J. Kigami, Localized eigenfunctions of the Laplacian on p. c. f. self-similar sets, J. London Math. Soc., 56 (1997), 320-332. MR1489140 (99b:35162)
    • (1997) J. London Math. Soc. , vol.56 , pp. 320-332
    • Barlow, M.T.1    Kigami, J.2
  • 4
    • 0000835875 scopus 로고
    • Brownian motion on the Sierpinski gasket
    • MR0966175 (89g:60241)
    • M.T. Barlow and E.A. Perkins, Brownian motion on the Sierpinski gasket. Probab. Theory Related Fields 79 (1988), 543-623. MR0966175 (89g:60241)
    • (1988) Probab. Theory Related Fields , vol.79 , pp. 543-623
    • Barlow, M.T.1    Perkins, E.A.2
  • 5
    • 84966250141 scopus 로고
    • Condensed Julia sets, with an application to a fractal lattice model Hamiltonian
    • MR0776392 (86h:58088)
    • M.F. Barnsley, J.S. Geronimo and A.N. Harrington, Condensed Julia sets, with an application to a fractal lattice model Hamiltonian. . Trans. Amer. Math. Soc. 288 (1985), 537-561. MR0776392 (86h:58088)
    • (1985) Trans. Amer. Math. Soc. , vol.288 , pp. 537-561
    • Barnsley, M.F.1    Geronimo, J.S.2    Harrington, A.N.3
  • 6
    • 0001511211 scopus 로고    scopus 로고
    • What is not in the domain of the Laplacian on Sierpinski gasket type fractals
    • MR1707752 (2001e:31016)
    • O. Ben-Bassat, R.S. Strichartz and A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals. J. Funct. Anal., 166 (1999), 197-217. MR1707752 (2001e:31016)
    • (1999) J. Funct. Anal. , vol.166 , pp. 197-217
    • Ben-Bassat, O.1    Strichartz, R.S.2    Teplyaev, A.3
  • 7
    • 0021206616 scopus 로고
    • Mellin transforms associated with Julia sets and physical applications
    • MR0739123 (85i:58067)
    • D. Bessis, J. Geronimo and P. Moussa, Mellin transforms associated with Julia sets and physical applications, J. Statist. Phys. 34 (1984), 75-110. MR0739123 (85i:58067)
    • (1984) J. Statist. Phys. , vol.34 , pp. 75-110
    • Bessis, D.1    Geronimo, J.2    Moussa, P.3
  • 8
    • 34250088878 scopus 로고
    • Function weighted measures and orthogonal polynomials on Julia sets
    • Erratum, Constr. Approx. 6 (1990), 335-336. MR0932652 (89h:58085); MR1054759 (91b:58210)
    • D. Bessis, J. Geronimo and P. Moussa, Function weighted measures and orthogonal polynomials on Julia sets, Constr. Approx. 4 (1988), 157-173. Erratum, Constr. Approx. 6 (1990), 335-336. MR0932652 (89h:58085); MR1054759 (91b:58210)
    • (1988) Constr. Approx. , vol.4 , pp. 157-173
    • Bessis, D.1    Geronimo, J.2    Moussa, P.3
  • 9
    • 33748428250 scopus 로고    scopus 로고
    • Fractal Laplacians on the unit interval
    • MR2103098 (2006b:34192)
    • E.J. Bird, S.-M. Ngai and A. Teplyaev, Fractal Laplacians on the Unit Interval, Ann. Sci. Math. Quebec 27 (2003), 135-168. MR2103098 (2006b:34192)
    • (2003) Ann. Sci. Math. Quebec , vol.27 , pp. 135-168
    • Bird S -M Ngai, E.J.1    Teplyaev, A.2
  • 10
    • 84950357804 scopus 로고
    • Invariant sets under iteration ofrational functions
    • MR0194595 (33:2805)
    • H. Brolin, Invariant sets under iteration ofrational functions, Ark. Mat. 6 (1965), 103-144. MR0194595 (33:2805)
    • (1965) Ark. Mat. , vol.6 , pp. 103-144
    • Brolin, H.1
  • 11
    • 0347917936 scopus 로고    scopus 로고
    • Fractal differential equations on the Sier-pinski gasket
    • MR1683211 (2000k:31016)
    • K. Dalrymple, R.S. Strichartz and J.P. Vinson, Fractal differential equations on the Sier-pinski gasket. J. Fourier Anal. Appl., 5 (1999), 203-284. MR1683211 (2000k:31016)
    • (1999) J. Fourier Anal. Appl. , vol.5 , pp. 203-284
    • Dalrymple, K.1    Strichartz, R.S.2    Vinson, J.P.3
  • 12
    • 77951030043 scopus 로고    scopus 로고
    • The zeta function of the Laplacian on certain fractals, to appear in Trans
    • G. Derfel, P. Grabner and F. Vogl, The zeta function of the Laplacian on certain fractals, to appear in Trans. Amer. Math. Soc.
    • Amer. Math. Soc.
    • Derfel, G.1    Grabner, P.2    Vogl, F.3
  • 13
    • 0000272107 scopus 로고
    • Solutions to the Schrodinger equation on some fractal lattices
    • MR0717348 (85h:82033)
    • E. Domany, S. Alexander, D. Bensimon and L. Kadanoff, Solutions to the Schrodinger equation on some fractal lattices. Phys. Rev. B (3) 28 (1984), 3110-3123. MR0717348 (85h:82033)
    • (1984) Phys. Rev. B (3) , vol.28 , pp. 3110-3123
    • Domany, E.1    Alexander, S.2    Bensimon, D.3    Kadanoff, L.4
  • 14
    • 0000888015 scopus 로고
    • Sur les iequations fonctionnelles
    • 48 (1920), 33-94, 208-314 MR1504787; MR1504792; MR1504797
    • P. Fatou, Sur les iequations fonctionnelles. Bull. Soc. Math. France, 47 (1919), 161-271; 48 (1920), 33-94, 208-314. MR1504787; MR1504792; MR1504797
    • (1919) Bull. Soc. Math. France , vol.47 , pp. 161-271
    • Fatou, P.1
  • 15
    • 0000648822 scopus 로고
    • Dirichlet forms, diffusion processes and spectral dimensions for nested fractals
    • Oslo, 1988 Cambridge Univ. Press, Cambridge MR1190496 (94d:60129)
    • M. Fukushima, Dirichlet forms, diffusion processes and spectral dimensions for nested fractals. Ideas and methods in Mathematical Analysis, Stochastics, and applications (Oslo, 1988), 151-161, Cambridge Univ. Press, Cambridge, 1992. MR1190496 (94d:60129)
    • (1992) Ideas and Methods in Mathematical Analysis, Stochastics, and Applications , pp. 151-161
    • Fukushima, M.1
  • 16
    • 0002832308 scopus 로고
    • On a spectral analysis for the Sierpinski gasket
    • MR1245223 (95b:31009)
    • M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket. Potential Analysis 1 (1992), 1-35. MR1245223 (95b:31009)
    • (1992) Potential Analysis , vol.1 , pp. 1-35
    • Fukushima, M.1    Shima, T.2
  • 17
    • 0001582569 scopus 로고
    • Random walks and diffusions on fractals
    • H. Kesten, ed. IMA Math. Appl. Springer, New York MR0894545 (88g:60245)
    • S. Goldstein, Random walks and diffusions on fractals, in "Percolation Theory and Ergodic Theory of Infinite Particle Systems" (H. Kesten, ed.), 121-129, IMA Math. Appl., Vol. 8, Springer, New York, 1987. MR0894545 (88g:60245)
    • (1987) Percolation Theory and Ergodic Theory of Infinite Particle Systems , vol.8 , pp. 121-129
    • Goldstein, S.1
  • 18
    • 0346446093 scopus 로고    scopus 로고
    • Functional iterations and stopping times for Brownian motion on the Sierpiniski gasket
    • MR1600494 (99b:60128)
    • P. Grabner, Functional iterations and stopping times for Brownian motion on the Sierpiniski gasket. Mathematika 44 (1997), 374-400. MR1600494 (99b:60128)
    • (1997) Mathematika , vol.44 , pp. 374-400
    • Grabner, P.1
  • 19
    • 77951516336 scopus 로고
    • A harmonic calculus on the Sierpinski spaces
    • MR1001286 (91g:31005)
    • J. Kigami, A harmonic calculus on the Sierpinski spaces. Japan J. Appl. Math. 6 (1989), 259-290. MR1001286 (91g:31005)
    • (1989) Japan J. Appl. Math. , vol.6 , pp. 259-290
    • Kigami, J.1
  • 20
    • 84924341601 scopus 로고
    • Harmonic calculus on p. c. f. self-similar sets
    • MR1076617 (93d:39008)
    • J. Kigami, Harmonic calculus on p. c. f. self-similar sets. Trans. Amer. Math. Soc. 335 (1993), 721-755. MR1076617 (93d:39008)
    • (1993) Trans. Amer. Math. Soc. , vol.335 , pp. 721-755
    • Kigami, J.1
  • 21
    • 0038404101 scopus 로고    scopus 로고
    • Analysis on fractals
    • Cambridge University Press MR1840042 (2002c:28015)
    • J. Kigami, Analysis on fractals. Cambridge Tracts in Mathematics 143, Cambridge University Press, 2001. MR1840042 (2002c:28015)
    • (2001) Cambridge Tracts in Mathematics , vol.143
    • Kigami, J.1
  • 22
    • 21344480546 scopus 로고
    • Weyl's problem for the spectral distribution of Lapla-cians on p. c. f. self-similar fractals
    • MR1243717 (94m:58225)
    • J. Kigami and M.L. Lapidus, Weyl's problem for the spectral distribution of Lapla-cians on p. c. f. self-similar fractals. Comm. Math. Phys. 158 (1993), 93-125. MR1243717 (94m:58225)
    • (1993) Comm. Math. Phys. , vol.158 , pp. 93-125
    • Kigami, J.1    Lapidus, M.L.2
  • 23
    • 0035529350 scopus 로고    scopus 로고
    • Self-similarity of volume measures for Laplacians on p. c. f. self-similar fractals
    • DOI 10.1007/s002200000326
    • J. Kigami and M.L. Lapidus, Self-similarity of volume measures for Laplacians on p. c. f. self-similar fractals. Comm. Math. Phys. 217 (2001), 165-180. MR1815029 (2002j:35237) (Pubitemid 33380265)
    • (2001) Communications in Mathematical Physics , vol.217 , Issue.1 , pp. 165-180
    • Kigami, J.1    Lapidus, M.L.2
  • 24
    • 21144471667 scopus 로고
    • Harmonization and homogenization on fractals
    • MR1218305 (94c:35026)
    • S. Kozlov, Harmonization and homogenization on fractals. Comm. Math. Phys. 153 (1993), 339-357. MR1218305 (94c:35026)
    • (1993) Comm. Math. Phys. , vol.153 , pp. 339-357
    • Kozlov, S.1
  • 25
    • 0742288933 scopus 로고    scopus 로고
    • Asymptotics of the transition probabilities of the simple random walk on self-similar graphs
    • MR2020038 (2004k:60130)
    • B. Kron and E. Teufl, Asymptotics of the transition probabilities of the simple random walk on self-similar graphs, Trans. Amer. Math. Soc., 356 (2003) 393-414. MR2020038 (2004k:60130)
    • (2003) Trans. Amer. Math. Soc. , vol.356 , pp. 393-414
    • Kron, B.1    Teufl, E.2
  • 26
    • 77957083459 scopus 로고
    • Spectral and fractal geometry: From the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function
    • (Birmingham, AL, 1990) Math. Sci. Engrg. Academic Press, Boston, MA MR1126694 (93f:58239)
    • M.L. Lapidus, Spectral and fractal geometry: From the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function. Differential equations and mathematical physics (Birmingham, AL, 1990), 151-181, Math. Sci. Engrg., 186, Academic Press, Boston, MA, 1992. MR1126694 (93f:58239)
    • (1992) Differential Equations and Mathematical Physics , vol.186 , pp. 151-181
    • Lapidus, M.L.1
  • 29
    • 0001893487 scopus 로고
    • Hypothe. se de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiee
    • MR1115940 (92f:11118)
    • M.L. Lapidus and H. Maier, Hypothe. se de Riemann, cordes fractales vibrantes et conjecture de Weyl-Berry modifiee. Acad. Sci. Paris Ser. I Math. 313, (1991), 19-24. MR1115940 (92f:11118)
    • (1991) Acad. Sci. Paris Ser. i Math. , vol.313 , pp. 19-24
    • Lapidus, M.L.1    Maier, H.2
  • 30
    • 0003176208 scopus 로고
    • The Riemann hypothesis and inverse spectral problems for fractal strings
    • MR1345711 (97b:11111)
    • M.L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings. J. London Math. Soc. (2) 52, (1995), 15-34. MR1345711 (97b:11111)
    • (1995) J. London Math. Soc. (2) , vol.52 , pp. 15-34
    • Lapidus, M.L.1    Maier, H.2
  • 31
    • 0000382625 scopus 로고
    • Fonction zeta de Riemann et conjecture de Weyl-Berry pour les tambours fractals
    • MR1046509 (91d:58248)
    • M.L. Lapidus and C. Pomerance, Fonction zeta de Riemann et conjecture de Weyl-Berry pour les tambours fractals. C.R. Acad. Sci. Paris Ser. I Math. 310, (1990), 343-348. MR1046509 (91d:58248)
    • (1990) C.R. Acad. Sci. Paris Ser. i Math. , vol.310 , pp. 343-348
    • Lapidus, M.L.1    Pomerance, C.2
  • 32
    • 84963042079 scopus 로고
    • The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums
    • MR1189091 (93k:58217)
    • M.L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. Proc. London Math. Soc. (3) 66, (1993), 41-69. MR1189091 (93k:58217)
    • (1993) Proc. London Math. Soc. (3) , vol.66 , pp. 41-69
    • Lapidus, M.L.1    Pomerance, C.2
  • 33
    • 1542273875 scopus 로고    scopus 로고
    • Self-Similarity, Operators and Dynamics
    • DOI 10.1023/A:1024931603110
    • L. Malozemov and A. Teplyaev, Self-similarity, operators and dynamics. Math. Phys. Anal. Geom. 6 (2003), 201-218. MR1997913 (2004d:47012) (Pubitemid 36921917)
    • (2003) MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY , vol.6 , Issue.3 , pp. 201-218
    • Malozemov, L.1    Teplyaev, A.2
  • 34
    • 0042387082 scopus 로고
    • Gaussian and non-Gaussian estimates for heat kernels on the Sierpiniski gasket
    • Beijing, 1993 de Gruyter, Berlin MR1366443 (97b:60132)
    • V. Metz and K.-T. Sturm, Gaussian and non-Gaussian estimates for heat kernels on the Sierpiniski gasket. Dirichlet forms and stochastic processes (Beijing, 1993), 283-289, de Gruyter, Berlin, 1995. MR1366443 (97b:60132)
    • (1995) Dirichlet Forms and Stochastic Processes , pp. 283-289
    • Metz, V.1    Sturm, K.-T.2
  • 35
    • 0021372307 scopus 로고
    • Spectrum of harmonic excitations on fractals
    • MR0737523 (85d:82101)
    • R. Rammal, Spectrum of harmonic excitations on fractals. J. Physique 45 (1984), 191-206. MR0737523 (85d:82101)
    • (1984) J. Physique , vol.45 , pp. 191-206
    • Rammal, R.1
  • 36
    • 0020498687 scopus 로고
    • Random walks on fractal structures and percolation clusters
    • R. Rammal and G. Toulouse, Random walks on fractal structures and percolation clusters. J. Physique Letters 44 (1983), L13-L22. (Pubitemid 13492296)
    • (1983) Journal de physique. Lettres , vol.44 , Issue.1 , pp. 13-22
    • Rammal, R.1    Toulouse, G.2
  • 37
    • 77951068981 scopus 로고
    • Electrical networks, symplectic reductions, and application to the renormalization map ofself-similar lattices
    • C. Sabot, Electrical networks, symplectic reductions, and application to the renormalization map ofself-similar lattices. J. Physique Letters 44 (1983), L13-L22.
    • (1983) J. Physique Letters , vol.44
    • Sabot, C.1
  • 38
    • 33845651625 scopus 로고    scopus 로고
    • Fractal geometry and applications: A Jubilee of Benoit Mandelbrot, Part 1
    • Amer. Math. Soc. MR2112106 (2005m:34202)
    • Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1. Proceedings of Symposia in Pure Mathematics 72, Amer. Math. Soc., (2004), 155-205. MR2112106 (2005m:34202)
    • (2004) Proceedings of Symposia in Pure Mathematics , vol.72 , pp. 155-205
  • 39
    • 0002230189 scopus 로고    scopus 로고
    • On eigenvalue problems for Laplacians on P.C.F. self-similar sets
    • T. Shima, On eigenvalue problems for Laplacians on p. c. f. self-similar sets, Japan J. Indust. Appl. Math. 13 (1996), 1-23. MR1377456 (97f:28028) (Pubitemid 126489600)
    • (1996) Japan Journal of Industrial and Applied Mathematics , vol.13 , Issue.1 , pp. 1-23
    • Shima, T.1
  • 40
    • 0000604124 scopus 로고    scopus 로고
    • Spectral analysis on infinite sierpiński gaskets
    • DOI 10.1006/jfan.1998.3297, PII S0022123698932979
    • A. Teplyaev, Spectral Analysis on Infinite Sierpiniski Gaskets, J. Funct. Anal., 159 (1998), 537-567. MR1658094 (99j:35153) (Pubitemid 128346255)
    • (1998) Journal of Functional Analysis , vol.159 , Issue.2 , pp. 537-567
    • Teplyaev, A.1
  • 41
    • 54949135857 scopus 로고    scopus 로고
    • Spectral zeta function of symmetric Sierpiniski gasket type fractals
    • Birkhaouser MR2087144 (2005h:28028)
    • A. Teplyaev, Spectral zeta function of symmetric Sierpiniski gasket type fractals, Fractal Geometry and Stochastics III, Progress in Probability 57, Birkhaouser (2004), 245-262. MR2087144 (2005h:28028)
    • (2004) Fractal Geometry and Stochastics III, Progress in Probability , vol.57 , pp. 245-262
    • Teplyaev, A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.