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Invariant measures for non-primitive tiling substitutions

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Abstract

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.

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Authors and Affiliations

  1. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Av. Libertador Bernardo O’Higgins, 3363, Santiago, Chile

    M. I. Cortez

  2. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    B. Solomyak

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  1. M. I. Cortez
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  2. B. Solomyak
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Correspondence to M. I. Cortez.

Additional information

Partially supported by Fondecyt 1100318

Partially supported by NSF grant DMS-0654408.

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Cortez, M.I., Solomyak, B. Invariant measures for non-primitive tiling substitutions. JAMA 115, 293–342 (2011). https://doi.org/10.1007/s11854-011-0031-x

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  • DOI: https://doi.org/10.1007/s11854-011-0031-x

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