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The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems

Rodrigo Treviño

Abstract

For typical properly ordered and minimal Bratteli diagrams $(B,\leq_r)$, it is shown that there are finitely many invariant distributions $\mathcal{D}_i$ which are the only obstructions to solving the cohomological equation $f = u-u\circ φ$ for the corresponding adic transformation $φ:X_B\rightarrow X_B$ and for $α$-Hölder $f$ with $α$ large enough. These invariant distributions are then used to define cyclic cocycles, a.k.a. traces $τ:K_0(\mathcal{A}_φ)\rightarrow \mathbb{R}$ for the crossed product algebra $\mathcal{A}_φ$.

The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems

Abstract

For typical properly ordered and minimal Bratteli diagrams , it is shown that there are finitely many invariant distributions which are the only obstructions to solving the cohomological equation for the corresponding adic transformation and for -Hölder with large enough. These invariant distributions are then used to define cyclic cocycles, a.k.a. traces for the crossed product algebra .

Paper Structure

This paper contains 10 sections, 12 theorems, 106 equations.

Table of Contents

  1. Introduction and statement of results
  2. Bratteli diagrams and minimal Cantor systems
  3. Bratteli diagrams
  4. Minimal Cantor systems
  5. Renormalization
  6. Inverse limits
  7. Function spaces
  8. The cohomological equation
  9. Tame estimates
  10. Cyclic cocycles

Key Result

Theorem 1.1

Let $\mu$ be a minimal, proper, $\sigma$-invariant ergodic probability measure on $\mathcal{O}_\beth$ satisfying Condition cond:1. Then there is a $d_\mu\in\mathbb{N}$ such that for $\mu$-almost every $x = (B,\leq_r)\in \mathcal{O}_\beth$, for any $\alpha>2$ there are $d_\mu$ obstructions to solving

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.1: Kesten-Furstenberg Theorem
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Definition 5.1
  • ...and 14 more