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Tree Capacity and Splitting Isometries for Subinvariant Kernels

James Tian

Abstract

Starting from a subinvariant positive definite kernel under a branching pullback, we attach to the resulting kernel tower a canonical electrical network on the word tree whose edge weights are the diagonal increments. This converts diagonal growth into effective resistance and capacity, giving explicit criteria and quantitative bounds, together with a matching upper bound under a mild level regularity condition. When the diagonal tower has finite limit at a point, we prove convergence of the full kernels and obtain an invariant completion with a minimality property. We also describe the associated RKHS splitting and a boundary martingale construction leading to weighted invariant majorants.

Tree Capacity and Splitting Isometries for Subinvariant Kernels

Abstract

Starting from a subinvariant positive definite kernel under a branching pullback, we attach to the resulting kernel tower a canonical electrical network on the word tree whose edge weights are the diagonal increments. This converts diagonal growth into effective resistance and capacity, giving explicit criteria and quantitative bounds, together with a matching upper bound under a mild level regularity condition. When the diagonal tower has finite limit at a point, we prove convergence of the full kernels and obtain an invariant completion with a minimality property. We also describe the associated RKHS splitting and a boundary martingale construction leading to weighted invariant majorants.
Paper Structure (7 sections, 38 theorems, 230 equations)

This paper contains 7 sections, 38 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Tree capacity/energy package
  3. Invariant completion
  4. Splitting isometry on $\mathcal{H}\left(K_{\infty}\right)$
  5. Boundary martingales
  6. Shift cocycles and diagonal boundary factors
  7. Weighted boundary kernels and invariant majorants

Key Result

Lemma 2.2

$L$ is order-preserving, i.e., $J_{1}\leq J_{2}\Rightarrow LJ_{1}\leq LJ_{2}$.

Theorems & Definitions (88)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Corollary 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 78 more