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Subshifts of finite symbolic rank

Su Gao, Ruiwen Li

TL;DR

This work investigates the relationship between symbolic rank and topological rank for Cantor minimal systems by introducing subshifts of finite symbolic rank as a natural analogue of finite rank measure-preserving transformations. It proves that minimal subshifts of finite symbolic rank have finite topological rank, and that every Cantor minimal system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank, aligning finite symbolic rank with finite alphabet rank via S-adic frameworks. The authors develop a robust framework using rank-n symbolic constructions, nested Kakutani–Rohlin partitions, Bratteli–Vershik representations, and S-adic subshifts to establish the main theorems, analyze descriptive complexity, and explore density, genericity, and factor properties. They also show that rank-1 systems are dense but not generic in several Polish coding spaces, while rank-2 subshifts are generic in spaces of transitive and totally transitive subshifts, and they characterize factor behavior, including the existence of infinite odometers as maximal equicontinuous factors for rank-2 subshifts.

Abstract

The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank-$1$ subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts we also show that the rank-1 subshifts are dense but not generic. Finally we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank $2$, and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.

Subshifts of finite symbolic rank

TL;DR

This work investigates the relationship between symbolic rank and topological rank for Cantor minimal systems by introducing subshifts of finite symbolic rank as a natural analogue of finite rank measure-preserving transformations. It proves that minimal subshifts of finite symbolic rank have finite topological rank, and that every Cantor minimal system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank, aligning finite symbolic rank with finite alphabet rank via S-adic frameworks. The authors develop a robust framework using rank-n symbolic constructions, nested Kakutani–Rohlin partitions, Bratteli–Vershik representations, and S-adic subshifts to establish the main theorems, analyze descriptive complexity, and explore density, genericity, and factor properties. They also show that rank-1 systems are dense but not generic in several Polish coding spaces, while rank-2 subshifts are generic in spaces of transitive and totally transitive subshifts, and they characterize factor behavior, including the existence of infinite odometers as maximal equicontinuous factors for rank-2 subshifts.

Abstract

The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank- subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts we also show that the rank-1 subshifts are dense but not generic. Finally we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank , and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.
Paper Structure (19 sections, 38 theorems, 64 equations)

This paper contains 19 sections, 38 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Descriptive set theory
  4. Topological dynamical systems
  5. Minimal Cantor systems
  6. Ordered Bratteli diagrams
  7. Kakutani--Rohlin partitions
  8. Subshifts
  9. Subshifts of finite symbolic rank
  10. Some computations of descriptive complexity
  11. A characterization of minimal rank-$1$ subshifts
  12. Proper finite rank constructions
  13. Finite symbolic rank and finite topological rank
  14. From finite symbolic rank to finite topological rank
  15. From finite topological rank to finite symbolic rank
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $(X, T)$ be an essentially minimal Cantor system, $Y$ be the unique minimal set, $y\in Y$ and $Z$ be a clopen subset of $X$ containing $y$, and let $\mathcal{Q}$ be a finite partition of $X$ into clopen sets. Then there is a Kakutani--Rohlin partition $\mathcal{P}$ such that $y\in B(\mathcal{P})

Theorems & Definitions (70)

  • Lemma 2.1: HPS, Lemma 4.1
  • Theorem 2.2: HPS, Theorem 4.2
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • ...and 60 more