Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T19:53:05.176Z Has data issue: true hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© Cambridge University Press, 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avila, A. and Delecroix, V.. Some monoids of Pisot matrices. Preprint, 2015, arXiv:1506.03692.Google Scholar
Akiyama, S. and Harriss, E.. Pentagonal domain exchange. Discrete Contin. Dyn. Syst. 33(10) (2013), 4375–4400.Google Scholar
Arnoux, P. and Ito, S.. Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8(2) (2001), 181–207. Journées Montoises d’Informatique Théorique (Marne-la-Vallée, 2000).Google Scholar
Adler, R., Kitchens, B. and Tresser, C.. Dynamics of non-ergodic piecewise affine maps of the torus. Ergod. Th. & Dynam. Sys. 21(4) (2001), 959–999.Google Scholar
Buzzi, J.. Piecewise isometries have zero topological entropy. Ergod. Th. & Dynam. Sys. 21(5) (2001), 1371–1377.Google Scholar
Goetz, A.. Piecewise isometries—an emerging area of dynamical systems. Fractals in Graz 2001 (Trends in Mathematics) . Birkhäuser, Basel, 2003, pp. 135–144.Google Scholar
Gowers, W. T.. Rough structure and classification. Geom. Funct. Anal. Special Volume(Part I) (2000), 79–117. GAFA 2000 (Tel Aviv, 1999).Google Scholar
Haller, H.. Rectangle exchange transformations. Monatsh. Math. 91(3) (1981), 215–232.Google Scholar
Hooper, W. P.. Renormalization of polygon exchange maps arising from corner percolation. Invent. Math. 191(2) (2013), 255–320.Google Scholar
Kenyon, R.. Self-replicating tilings. Symbolic Dynamics and Its Applications (New Haven, CT, 1991) (Contemporary Mathematics, 135) . American Mathematical Society, Providence, RI, 1992, pp. 239–263.Google Scholar
Kenyon, R.. The construction of self-similar tilings. Geom. Funct. Anal. 6(3) (1996), 471–488.Google Scholar
Lagarias, J. C.. Meyer’s concept of quasicrystal and quasiregular sets. Comm. Math. Phys. 179(2) (1996), 365–376.Google Scholar
Lind, D. A.. The entropies of topological Markov shifts and a related class of algebraic integers. Ergod. Th. & Dynam. Sys. 4(2) (1984), 283–300.Google Scholar
Lowenstein, J. H., Kouptsov, K. L. and Vivaldi, F.. Recursive tiling and geometry of piecewise rotations by 𝜋/7. Nonlinearity 17(2) (2004), 371–395.Google Scholar
Meyer, Y.. Quasicrystals, Diophantine approximation and algebraic numbers. Beyond Quasicrystals (Les Houches, 1994). Springer, Berlin, 1995, pp. 3–16.Google Scholar
Poincaré, H.. The Three-body Problem and the Equations of Dynamics (Astrophysics and Space Science Library, 443) . Springer, Cham, 2017. Poincaré’s foundational work on dynamical systems theory. Translated from the 1890 French original and with a preface by Bruce D. Popp.Google Scholar
Rauzy, G.. Échanges d’intervalles et transformations induites. Acta Arith. 34(4) (1979), 315–328.Google Scholar
Rauzy, G.. Nombres algébriques et substitutions. Bull. Soc. Math. France 110(2) (1982), 147–178.Google Scholar
Schwartz, R. E.. The Octogonal PETs (Mathematical Surveys and Monographs, 197) . American Mathematical Society, Providence, RI, 2014.Google Scholar
Stein, E. M. and Shakarchi, R.. Fourier Analysis (Princeton Lectures in Analysis, 1) . Princeton University Press, Princeton, NJ, 2003, An introduction.Google Scholar
Thurston, W. P.. Entropy in dimension one. Frontiers in Complex Dynamics (Princeton Mathematics Series, 51) . Princeton University Press, Princeton, NJ, 2014, pp. 339–384.Google Scholar