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A family of minimal and renormalizable rectangle exchange maps

Published online by Cambridge University Press:  27 November 2019

IAN ALEVY Open the ORCID record for IAN ALEVY [Opens in a new window] ,
RICHARD KENYON  and
REN YI
IAN ALEVY
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY14627, USA email ian.alevy@rochester.edu
RICHARD KENYON
Affiliation:
Department of Mathematics, Yale University, New Haven, CT06520, USA email richard.kenyon@yale.edu
REN YI
Affiliation:
Department of Mathematics, Brown University, Providence, RI02912, USA email ren_yi_1@alumni.brown.edu
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Abstract

A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a Pisot–Vijayaraghavan (PV) number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.

Keywords

low-dimensional dynamicssymbolic dynamicstilings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 3 , March 2021 , pp. 790 - 817
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Copyright
© Cambridge University Press, 2019

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