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Vladimirov-Pearson Operators on $ζ$-regular Ultrametric Cantor Sets

Patrick Erik Bradley

TL;DR

The paper addresses diffusion-type operators on zeta-regular ultrametric Cantor sets by constructing the Vladimirov-Pearson operator D^s from the Connes measure associated with the spectral triple and the zeta-function zeta(s). It shows D^s is an integral operator analogous to the Vladimirov-Taibleson operator on p-adic spaces, with a Markov semigroup, heat kernel representation, and a complete eigenbasis given by ultrametric wavelets whose eigenvalues are explicitly described. Key contributions include self-adjointness and positive semi-definiteness of D^s, a dichotomy in the spectrum depending on s (bounded/compact for s ≥ 4 and pure point for s < 4), and explicit kernel and Green function representations via the ultrametric wavelet basis. The results extend p-adic diffusion theory to ζ-regular ultrametric Cantor sets and provide a Sobolev-space framework for diffusion and potential theory on fractal ultrametric spaces, with explicit heat-kernel and Green-function formulas.

Abstract

A new operator for certain types of ultrametric Cantor sets is constructed using the measure coming from the spectral triple associated with the Cantor set, as well as its zeta function. Under certain mild conditions on that measure, it is shown that it is an integral operator similar to the Vladimirov-Taibleson operator on the p-adic integers. Its spectral properties are studied, and the Markov property and kernel representation of the heat kernel generated by this so-called \emph{Vladimirov-Pearson} operator is shown, viewed as acting on a certain Sobolev space. A large class of these operators have a heat kernel and a Green function explicitly given by the ultrametric wavelets on the Cantor set, which are eigenfunctions of the operator.

Vladimirov-Pearson Operators on $ζ$-regular Ultrametric Cantor Sets

TL;DR

The paper addresses diffusion-type operators on zeta-regular ultrametric Cantor sets by constructing the Vladimirov-Pearson operator D^s from the Connes measure associated with the spectral triple and the zeta-function zeta(s). It shows D^s is an integral operator analogous to the Vladimirov-Taibleson operator on p-adic spaces, with a Markov semigroup, heat kernel representation, and a complete eigenbasis given by ultrametric wavelets whose eigenvalues are explicitly described. Key contributions include self-adjointness and positive semi-definiteness of D^s, a dichotomy in the spectrum depending on s (bounded/compact for s ≥ 4 and pure point for s < 4), and explicit kernel and Green function representations via the ultrametric wavelet basis. The results extend p-adic diffusion theory to ζ-regular ultrametric Cantor sets and provide a Sobolev-space framework for diffusion and potential theory on fractal ultrametric spaces, with explicit heat-kernel and Green-function formulas.

Abstract

A new operator for certain types of ultrametric Cantor sets is constructed using the measure coming from the spectral triple associated with the Cantor set, as well as its zeta function. Under certain mild conditions on that measure, it is shown that it is an integral operator similar to the Vladimirov-Taibleson operator on the p-adic integers. Its spectral properties are studied, and the Markov property and kernel representation of the heat kernel generated by this so-called \emph{Vladimirov-Pearson} operator is shown, viewed as acting on a certain Sobolev space. A large class of these operators have a heat kernel and a Green function explicitly given by the ultrametric wavelets on the Cantor set, which are eigenfunctions of the operator.
Paper Structure (4 sections, 9 theorems, 56 equations)

This paper contains 4 sections, 9 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Regular Ultrametric Cantor sets
  3. Vladimir-Pearson operators
  4. Vladimirov-Pearson semigroup

Key Result

Theorem 3.7

Let $(C,d)$ be an equitising $\zeta$-regular Cantor set. Then $\mu_\tau$ is an equity measure, and the Vladimirov-Pearson operator on the space of test functions $\mathcal{D}(C)$ has the form for $s\in\mathds{R}$, where $a\wedge b$ is the join of $a,b\in C$ in the Michon tree $T(C)$, $\mathop{\mathrm{ch}}\nolimits(v)$ the set of child nodes of vertex $v\in V$, and $S_\epsilon(x)$ the sphere of ra

Theorems & Definitions (31)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Definition 3.1
  • Definition 3.2
  • Example 3.3: $p$-adic integers
  • Example 3.4: Level-regularity
  • ...and 21 more