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Luke G. Rogers

Publications and source records attributed to Luke G. Rogers.

25 records · Page 2Linked to original sources
arXiv2012

Derivations and Dirichlet forms on fractals

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, and refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.

math.OA↗
arXiv2010

Estimates for the resolvent kernel of the Laplacian on p.c.f. self similar fractals and blowups

We provide a method for obtaining upper estimates of the resolvent kernel of the Laplacian on a post-critically finite self-similar fractal that relies on a self-similar series decomposition of the resolvent. Decay estimates on the positive real axis are proved by analyzing functions satisfying an interior eigenfunction condition with positive eigenvalue. These lead to estimates on the complement of the negative real axis via the Phragmen-Lindelof theorem. Applications are given to kernels for functions of the Laplacian, including the heat kernel, and to proving the existence of a self-similar series decomposition for the Laplacian resolvent on fractal blowups.

math.CA↗
arXiv2009

The resolvent kernel for PCF self-similar fractals

For the Laplacian $Δ$ defined on a p.c.f. self-similar fractal, we give an explicit formula for the resolvent kernel of the Laplacian with Dirichlet boundary conditions, and also with Neumann boundary conditions. That is, we construct a symmetric function $G^{(λ)}$ which solves $(λ\mathbb{I} - Δ)^{-1} f(x) = \int G^{(λ)}(x,y) f(y) dμ(y)$. The method is similar to Kigami's construction of the Green kernel in \cite[\S3.5]{Kig01} and is expressed as a sum of scaled and "translated" copies of a certain function $ψ^{(λ)}$ which may be considered as a fundamental solution of the resolvent equation. Examples of the explicit resolvent kernel formula are given for the unit interval, standard Sierpinski gasket, and the level-3 Sierpinski gasket $SG_3$.

math.AP↗
arXiv2009

Distribution theory on p.c.f. fractals

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested.

math.FA↗
arXiv2008

Szego limit theorems on the Sierpinski gasket

We use the existence of localized eigenfunctions of the Laplacian on the Sierpinski gasket to formulate and prove analogues of the strong Szego limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences.

math.SP↗
arXiv2008

Laplacians on the basilica Julia set

We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is 2, the spectral dimension is $\log9/\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\mapsto P(z)$; it has energy renormalization factor $\sqrt2$ and spectral dimension 4/3, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set.

math.CA↗