[Paper Review] Fusion: a general framework for hierarchical tilings of $R^d$

Natalie Priebe Frank, Lorenzo Sadun|arXiv (Cornell University)|Jan 25, 2011

Mathematical Dynamics and Fractals49 references17 citations

TL;DR

This paper introduces Fusion, a general framework for hierarchical tilings in ℝ^d that unifies substitution tilings, cut-and-stack systems, S-adic transformations, and Bratteli-Vershik systems. It establishes conditions under which key dynamical properties—such as unique ergodicity, pure point spectrum, and topological mixing—carry over from substitution systems, while providing counterexamples showing these properties can fail without strong assumptions, including a 2D tiling space with pure point measure-theoretic spectrum but topological weak mixing.

ABSTRACT

We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing.

Motivation & Objective

• To develop a general formalism for hierarchical tilings in ℝ^d that unifies diverse systems like substitution tilings, cut-and-stack, S-adic, and Bratteli-Vershik systems.
• To identify minimal assumptions under which classical properties of substitution tilings—such as unique ergodicity and pure point spectrum—extend to general fusion tilings.
• To construct counterexamples demonstrating that properties like unique ergodicity and topological mixing can fail without strong assumptions, such as primitivity or recognizability.
• To explore the interplay between combinatorial complexity, cohomology, and dynamical behavior in fusion tiling spaces, especially in non-Pisot and non-asymptotically FLC cases.
• To extend tools like Anderson-Putnam and Barge-Diamond complexes to non-substitution fusion systems and investigate their cohomological structure.

Proposed method

• Define fusion tilings via transition matrices and subdivision maps, generalizing substitution rules to allow non-uniform hierarchical growth.
• Introduce key assumptions: prototile- and transition-regularity, primitivity, recognizability, and van Hove sequences to control asymptotic behavior.
• Use inverse limit structures and collaring techniques (Anderson-Putnam and Barge-Diamond) to analyze tiling cohomology and control patch structure.
• Apply Perron-Frobenius theory and spectral analysis to study measurable and topological eigenvalues, distinguishing between pure point, mixed, and continuous spectrum.
• Construct explicit examples—such as the Fibonacci and non-Pisot DPV systems—to test the necessity of assumptions and reveal dynamical pathologies.
• Analyze asymptotic self-similarity and asymptotic FLC (finite local complexity) to characterize long-range geometric and combinatorial behavior of supertiles.

Experimental results

Research questions

• RQ1 Under what conditions do substitution-like properties—such as pure point spectrum or unique ergodicity—extend to general fusion tilings?
• RQ2 Can a tiling space be topologically weakly mixing yet have pure point measure-theoretic spectrum, and if so, how is this possible?
• RQ3 How does the failure of primitivity or recognizability affect the existence and structure of invariant measures and spectral properties?
• RQ4 What is the role of the largest eigenvalue of the transition matrix in determining the asymptotic geometry and combinatorial complexity of fusion tilings?
• RQ5 How do cohomological invariants like Čech cohomology groups Ȟ¹ and Ȟ² behave in fusion systems that are not substitution-based?

Key findings

• A minimal tiling space exists that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another having mixed spectrum.
• A 2-dimensional tiling space is constructed with pure point measure-theoretic spectrum but is topologically weakly mixing, demonstrating a separation between spectral and topological mixing properties.
• The Fibonacci DPV fusion system has finitely generated Ȟ¹ and Ȟ², with rank 64 for Ȟ², and is asymptotically FLC and asymptotically self-similar with limiting prototile ratios φ:1.
• The non-Pisot DPV system has transition matrix with largest eigenvalue ((1+√13)/2)², not a Pisot number, leading to unbounded growth in adjacency types and infinitely generated Ȟ¹ and Ȟ².
• The non-Pisot system is not asymptotically FLC, and changes in tile sizes under fixed fusion rules can alter the dynamics and even the topology of the tiling space.
• Asymptotically self-affine fusion rules exist where the limit of scaled supertiles exists, but such systems may fail to be asymptotically FLC, implying the limiting self-affine tiling has infinite local complexity.