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SUMMARY:The Laplacian on some finitely ramified self-conformal circle pack
 ing fractals and Weyl's asymptotics for its eigenvalues - Naotaka Kajino (
 Kobe University\, Japan)
DTSTART:20170704T151500Z
DTEND:20170704T161500Z
UID:TALK73170@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:The purpose of this talk is to present the speaker's recent re
 search\nin progress on the construction of a ``canonical'' Laplacian on fi
 nitely\nramified circle packing fractals invariant with respect to a famil
 y of\nMoebius transformations and on Weyl's asymptotics for its eigenvalue
 s.\n\nIn the simplest case of the Apollonian gasket\, the speaker has obta
 ined\nan explicit expression of a certain canonical Dirichlet form in term
 s of the\ncircle packing structure of the fractal. Our Laplacian on a gene
 ral circle\npacking fractal is constructed by adopting the same kind of ex
 pression\nas the definition of a (seemingly canonical) strongly local Diri
 chlet form.\nWeyl's eigenvalue asymptotics for this Laplacian has been als
 o established\nin some important examples including the Apollonian gasket\
 , and the proof\nof this result heavily relies on ergodic-theoretic analys
 is of a Markov\nchain on\nthe space of ``shapes of cells'' resulting from 
 a suitable cellular\ndecomposition\nof the fractal.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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