Finite and infinite invariant measures for adic transformations

TL;DR

This work extends the invariant-measure classification for Vershik adic transformations from stationary (and primitive) systems to nonstationary Bratteli diagrams of bounded rank, accommodating measures finite on a subdiagram yet possibly infinite on the ambient space. Central to the approach are a nonstationary Frobenius decomposition, a nonstationary Frobenius–Victory theorem, and the construction of adic towers with canonical covers, which together translate invariant measures into distinguished eigenvector sequences across primitive diagonal blocks. The authors prove necessary and sufficient criteria distinguishing finite from infinite invariant measures, including a complete description for measures finite on some subdiagram, and show how tower measures can be finite or infinite depending on the distinguished-status of block-eigenvectors. The framework yields concrete models (e.g., nested circle rotations and the Integer Cantor Set) and reveals how subdiagrams generate new invariant towers with masses that may be finite or infinite, enriching infinite-ergodic theory. Overall, the paper provides a robust, general machinery to classify FC-invariant Borel measures in nonstationary adic settings, connecting tower- and cover- constructions with subdiagrams and extending the reach of stationary BKMS-type results to a broad nonstationary regime with locally finite and locally infinite phenomena.

Abstract

We classify the invariant Borel measures for adic transformations, where the alphabets have bounded size and the measure is finite on the path space of some sub-Bratteli diagram. We develop a nonstationary version of the Frobenius normal form for a reducible matrix, present an appropriate nonstationary notion of distinguished eigenvector, and prove a nonstationary Frobenius--Victory theorem. This parallels the approach to the stationary case developed by Bezuglyi, Kwiatkowski, Medynets and Solomyak in arXiv:0812.1088 where they classify the locally finite invariant measures. In later work, they also address the nonstationary case. We extend this in two ways. Firstly, in both the stationary and nonstationary settings, we allow for measures which are locally infinite, motivating this extension with examples. Secondly we give a complete classification, presenting a necessary and sufficient condition for a measure which is finite on some subdiagram to be infinite for the original diagram. As part of our program, we introduce a related object called an adic tower and a construction called the canonical cover of the subdiagram: if the measure is finite on the subdiagram but locally infinite on the original space it will be locally finite on the cover space, though the path spaces for the original diagram and the cover are measure-theoretically isomorphic. Our examples include two new models for the Integer Cantor Set transformation of [Fisher1992], one locally finite and one locally infinite, which lead to a new class of examples related to fractal sets of integers: nested circle rotations, where one rotation is embedded in another. The resulting tower measure inside the original rotation can have finite or infinite total mass as determined by our general criterion.

Figures (9)

• Figure 1: Bratteli diagram for the Chacon adic transformation: the substitution $\rho(0)= 0$, $\rho(1)= 1101$; in this figure, arrows depict the direction of the substitution map, while the edge path arrows, indicating the direction of time, point in the opposite sense. The substitution corresponds to the edge order $a<b<c<d$; edge $e$ is ordered trivially as it is the only incoming edge to vertex $0$.
• Figure 2: Stable tree picture showing simultaneously points of the Chacon adic transformation and substitution dynamical system, indicating the Livshits factor map: the substitution maps downwards, vertex and edge paths are vertical, while a symbol string from the substitution dynamical system at with time $0$ is horizontal along the lowest row.
• Figure 3: A stable equivalence class of the Integer Cantor Set transformation, depicted in the curtain and stable tree models; $(x_i) _{i\in\mathbb N}$ is a vertex path for the Bratteli diagram while $(a_i) _{i\in\mathbb Z}$ is a point in the substitution dynamical system.
• Figure 4: The Integer Cantor Set as a tower inside the 3-adic odometer, and its canonical cover with substitution sequence $\rho_i\equiv\rho$.
• Figure 5: Symbol splitting: $A_k B_k = M_k$ and $B_k A_{k+1}=L_k$

Theorems & Definitions (194)

• Definition 1.1
• Remark 1.2
• Remark 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Example 2.4
• Definition 2.5
• Lemma 2.6
• proof