Geometriae Dedicata

, Volume 171, Issue 1, pp 149–186 | Cite as

Fusion: a general framework for hierarchical tilings of \(\mathbb{R }^d\)

  • Natalie Priebe Frank
  • Lorenzo Sadun
Original Paper


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.


Self-similar Substitution Mixing Dynamical spectrum Invariant measures 

Mathematics Subject Classification (1991)

Primary: 37B50 Secondary: 52C23 37A25 37B10 



We thank Mike Boyle, Lewis Bowen, Kariane Calta, Amos Nevo, E. Arthur Robinson, Jr. and Boris Solomyak for helpful discussions. The work of L.S. is partially supported by NSF Grants DMS-0701055 and DMS-1101326.


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsVassar CollegePoughkeepsieUSA
  2. 2.Department of MathematicsThe University of Texas at AustinAustinUSA

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