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Homoclinically Expansive Actions and a Garden of Eden Theorem for Harmonic Models

  • Tullio Ceccherini-SilbersteinEmail author
  • Michel Coornaert
  • Hanfeng Li
Article
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Abstract

Let \({\Gamma}\) be a countable Abelian group and \({f \in \mathbb{Z}[\Gamma]}\), where \({\mathbb{Z}[\Gamma]}\) denotes the integral group ring of \({\Gamma}\). Consider the Pontryagin dual Xf of the cyclic \({\mathbb{Z}[\Gamma]}\)-module \({\mathbb{Z}[\Gamma]/\mathbb{Z}[\Gamma] f}\) and suppose that f is weakly expansive (e.g., f is invertible in \({\ell^1(\Gamma)}\), or, when \({\Gamma}\) is not virtually \({\mathbb{Z}}\) or \({\mathbb{Z}^2}\), f is well-balanced) and that Xf is connected. We prove that if \({\tau \colon X_f \to X_f}\) is a \({\Gamma}\)-equivariant continuous map, then \({\tau}\) is surjective if and only if the restriction of \({\tau}\) to each \({\Gamma}\)-homoclinicity class is injective. We also show that this equivalence remains valid in the case when \({\Gamma = \mathbb{Z}^d}\) and \({f \in \mathbb{Z}[\Gamma] = \mathbb{Z}[u_1,u_1^{-1}, \ldots, u_d, u_d^{-1}]}\) is an irreducible atoral polynomial whose zero-set Z(f) satisfies some suitable finiteness conditions (e.g., when \({d \geq 2}\) such that Z(f) is finite). These two results are analogues of the classical Garden of Eden theorem of Moore and Myhill for cellular automata with finite alphabet over \({\Gamma}\).

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Notes

Acknowledgements

We express our gratitude to the referees for their careful reading of the paper and for providing useful and interesting comments that helped us improving our presentation.Hanfeng Liwas partially supported by NSF and NSFC grants.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di IngegneriaUniversità del SannioBeneventoItaly
  2. 2.Université de Strasbourg, CNRS, IRMA UMR 7501StrasbourgFrance
  3. 3.Department of MathematicsChongqing UniversityChongqingChina
  4. 4.Department of MathematicsSUNY at BuffaloBuffaloUSA

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