[Paper Review] $Z^{d}$-odometers and cohomology

Thierry Giordano, Ian F. Putnam+1 more|arXiv (Cornell University)|Sep 25, 2017

Advanced Operator Algebra Research4 citations

TL;DR

This paper investigates the cohomological invariants of minimal, free ℤ^d-odometers on the Cantor set, showing that the first cohomology group H¹ provides a complete conjugacy invariant for such systems—contrasting with orbit equivalence, where cohomology in dimension d is the relevant invariant. The work establishes a complete classification of ℤ^d-odometers up to conjugacy, isomorphism, orbit equivalence, and continuous orbit equivalence using cohomological and dynamical techniques.

ABSTRACT

Cohomology for actions of free abelian groups on the Cantor set has (when endowed with an order structure) provided a complete invariance for orbit equivalence. In this paper, we study a particular class of actions of such groups called odometers (or profinite actions) and investigate their cohomology. We show that for a free, minimal $\Z^{d}$-odometer, the first cohomology group provides a complete invariant for the action up to conjugacy. This is in contrast with the situation for orbit equivalence where it is the cohomology in dimension $d$ which provides the invariant. We also consider classification up to isomorphism and continuous orbit equivalence.

Motivation & Objective

• To classify ℤ^d-odometers on the Cantor set up to conjugacy, isomorphism, orbit equivalence, and continuous orbit equivalence.
• To investigate the role of cohomology in classifying these systems, particularly the first cohomology group H¹.
• To clarify the distinction between conjugacy invariants (H¹) and orbit equivalence invariants (H^d), resolving a key contrast in the theory.
• To establish that the maximal totally disconnected equicontinuous factor of a minimal ℤ^d-action arises from the image of H¹ under a specific map.
• To prove that the cohomological data determines the system up to conjugacy, using eigenfunctions and character theory on the equicontinuous factor.

Proposed method

• Use the maximal equicontinuous factor (X_eq, φ_eq) as a central object, constructed via the orbit closure of continuous eigenfunctions.
• Apply the duality between characters on the equicontinuous factor and cohomology classes via the map τ_μ¹: ℚ(H¹(X,φ)) → Hom(ℤ^d, 𝕋).
• Characterize continuous eigenfunctions as those factoring through the quotient X_eq / X_eq⁰, where X_eq⁰ is the connected component of the identity.
• Show that the image of τ_μ¹ identifies with the dual of X_eq / X_eq⁰, establishing a conjugacy between the original system and a group rotation system.
• Use the fact that minimal equicontinuous actions of ℤ^d are group rotations, enabling the construction of characters and the identification of finite-range eigenfunctions.
• Employ orbit cocycles and continuity conditions to analyze continuous orbit equivalence, distinguishing it from general orbit equivalence.

Experimental results

Research questions

• RQ1 How does the first cohomology group H¹(ℤ^d, X) classify ℤ^d-odometers up to conjugacy?
• RQ2 Why does H¹ serve as a complete invariant for conjugacy, while H^d is the invariant for orbit equivalence in this context?
• RQ3 What is the structure of the maximal totally disconnected equicontinuous factor of a minimal ℤ^d-action on the Cantor set?
• RQ4 How do continuous eigenfunctions and characters on the equicontinuous factor relate to cohomological data?
• RQ5 To what extent do continuous orbit equivalence and conjugacy differ in the setting of ℤ^d-odometers?

Key findings

• For a minimal, free ℤ^d-odometer on the Cantor set, the first cohomology group H¹ provides a complete invariant for conjugacy.
• The maximal equicontinuous factor (X_eq, φ_eq) is isomorphic to a group rotation on a compact abelian group, and its totally disconnected quotient X_eq / X_eq⁰ captures the cohomological data.
• The image of the map τ_μ¹: ℚ(H¹(X,φ)) → Hom(ℤ^d, 𝕋) identifies with the dual of X_eq / X_eq⁰, and this dual is isomorphic to H / ℤ^d where H = τ_μ¹(ℚ(H¹(X,φ))).
• The factor map π: (X,φ) → (Y_H, ψ_H) is the maximal totally disconnected equicontinuous factor, and (Y_H, ψ_H) is a group rotation system.
• The system (X,φ) is conjugate to (Y_H, ψ_H) if and only if the cohomological data τ_μ¹ identifies the group structure via the dual of X_eq / X_eq⁰.
• The cohomological classification via H¹ contrasts sharply with orbit equivalence, where the relevant invariant is H^d, highlighting a fundamental difference in invariants across equivalence relations.