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A general framework for quasi-isometries in symbolic dynamics beyond groups

Sebastián Barbieri, Nicolás Bitar

TL;DR

The paper introduces blueprints as a versatile algebraic framework that encodes families of countable graphs and enables symbolic dynamics under partial monoid actions. It develops a general transfer mechanism for invariants through quasi-isometries, extending results previously known for groups to broader structures, and proves that domino problem undecidability, the existence of strongly aperiodic SFTs, and Medvedev-degrees invariants persist under quasi-isometries of finitely presented blueprints. A central methodological innovation is the construction of a QI-encoding SFT, which binds quasi-isometries to transfers of subshifts, ensuring that dynamical properties survive across quasi-isometric presentations. The approach is then applied to d-dimensional geometric tilings, where a patch blueprint captures punctured tilings with finite local complexity, culminating in an undecidability result for the geometric domino problem in dimensions $d\ge 2$. Overall, the work provides a unifying, geometry-aware framework that broadens the reach of quasi-isometry rigidity in symbolic dynamics beyond groups, with concrete consequences for tiling problems and computability invariants.

Abstract

We introduce an algebraic structure which encodes a collection of countable graphs through a set of states, generators and relations. For these structures, which we call blueprints, we provide a general framework for symbolic dynamics under a partial monoid action, and for transferring invariants of their symbolic dynamics through quasi-isometries. In particular, we show that the undecidability of the domino problem, the existence of strongly aperiodic subshifts of finite type, and the existence of subshifts of finite type without computable points are all quasi-isometry invariants for finitely presented blueprints. As an application of this model, we show that a variant of the domino problem for geometric tilings of $\mathbb{R}^d$ is undecidable for $d \geq 2$ on any underlying tiling space with finite local complexity.

A general framework for quasi-isometries in symbolic dynamics beyond groups

TL;DR

The paper introduces blueprints as a versatile algebraic framework that encodes families of countable graphs and enables symbolic dynamics under partial monoid actions. It develops a general transfer mechanism for invariants through quasi-isometries, extending results previously known for groups to broader structures, and proves that domino problem undecidability, the existence of strongly aperiodic SFTs, and Medvedev-degrees invariants persist under quasi-isometries of finitely presented blueprints. A central methodological innovation is the construction of a QI-encoding SFT, which binds quasi-isometries to transfers of subshifts, ensuring that dynamical properties survive across quasi-isometric presentations. The approach is then applied to d-dimensional geometric tilings, where a patch blueprint captures punctured tilings with finite local complexity, culminating in an undecidability result for the geometric domino problem in dimensions . Overall, the work provides a unifying, geometry-aware framework that broadens the reach of quasi-isometry rigidity in symbolic dynamics beyond groups, with concrete consequences for tiling problems and computability invariants.

Abstract

We introduce an algebraic structure which encodes a collection of countable graphs through a set of states, generators and relations. For these structures, which we call blueprints, we provide a general framework for symbolic dynamics under a partial monoid action, and for transferring invariants of their symbolic dynamics through quasi-isometries. In particular, we show that the undecidability of the domino problem, the existence of strongly aperiodic subshifts of finite type, and the existence of subshifts of finite type without computable points are all quasi-isometry invariants for finitely presented blueprints. As an application of this model, we show that a variant of the domino problem for geometric tilings of is undecidable for on any underlying tiling space with finite local complexity.

Paper Structure

This paper contains 20 sections, 28 theorems, 117 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Structure of the article
  4. Blueprints
  5. Quotients of a blueprints
  6. Topology and dynamics of the model space
  7. Subshifts on Blueprints
  8. Quasi-isometries between finitely presented blueprints
  9. Quasi-metrics and Quasi-isometries
  10. Encoding quasi-isometries through subshifts
  11. QI-rigidity
  12. Domino problems for blueprints and models
  13. Strong aperiodicity
  14. Medvedev degrees
  15. A hyperbolic example
  16. ...and 5 more sections

Key Result

Theorem A

Let $\Gamma_1$, $\Gamma_2$ be two finitely presented strongly connected blueprints that are quasi-isometric. Then, the $\Gamma_1$-domino problem is decidable if and only if the $\Gamma_2$-domino problem is decidable.

Figures (7)

  • Figure 1: The graph of a model in the $1$-$2$ tree blueprint. The state $\texttt{0}$ is always followed by the single generator $s$ while the state $\texttt{1}$ is followed by the two generators $u,t$.
  • Figure 2: A portion of a model graph of $\mathcal{H}$
  • Figure 3: A portion of the hard-square subshift on a model graph of the hyperbolic tiling blueprint $\mathcal{H}$.
  • Figure 4: The hat tile
  • Figure 5: A patch generated by the punctured tiles of the monotile.
  • ...and 2 more figures

Theorems & Definitions (101)

  • Theorem A: Theorem \ref{['thm:domino']}
  • Theorem B: Theorem \ref{['thm:SA']}
  • Theorem C: Theorem \ref{['thm:domino_geom']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • ...and 91 more