The direct product of two Fibonacci tilings can be described as a genuine stone inflation rule with four prototiles. This rule admits various modifications, which lead to 48 different inflation rules, known as the direct product variations. They all result in tilings that are measure-theoretically isomorphic by the Halmos–von Neumann theorem. They can be described as cut and project sets with characteristic windows in a two-dimensional Euclidean internal space. Here, we analyze and classify them further, in particular, with respect to topological conjugacy.
REFERENCES
1.
M.
Baake
and U.
Grimm
, Aperiodic Order. Vol. 1: A. Mathematical Invitation
(Cambridge University Press
, Cambridge
, 2013
).2.
M.
Baake
, N. P.
Frank
, and U.
Grimm
, “Three variations on a theme by Fibonacci
,” Stochastics Dyn.
21
, 2140001
(2021
); arXiv:1910.00988.3.
B.
Sing
, “Pisot substitutions and beyond
,” Ph.D. thesis, Bielefeld University
, 2007
, available electronically at http://nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:hbz:361-11555.4.
K. R.
Wicks
, Fractals and Hypersurfaces
, Lecture Notes in Mathematics Vol. 1492 (Springer
, Berlin
, 1991
).5.
N. P.
Frank
, “A primer of substitution tilings of the Euclidean plane
,” Expositiones Math.
26
, 295
–326
(2008
); arXiv:0705.1142.6.
N. P.
Frank
and E. A.
Robinson
, “Generalized β-expansions, substitution tilings, and local finiteness
,” Trans. Am. Math. Soc.
360
, 1163
–1177
(2008
); arXiv:math.DS/0506098.7.
M.
Baake
and D.
Lenz
, “Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra
,” Ergodic Theory Dyn. Syst.
24
, 1867
–1893
(2004
).8.
M.
Baake
, M.
Schlottmann
, and P. D.
Jarvis
, “Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability
,” J. Phys. A: Math. Gen.
24
, 4637
–4654
(1991
).9.
D.
Frettlöh
, “More inflation tilings
,” in Aperiodic Order. Vol. 2: Crystallography and Almost Periodicity
, edited by M.
Baake
and U.
Grimm
(Cambridge University Press
, Cambridge
, 2017
), pp. 1
–35
.10.
B.
Solomyak
, “Dynamics of self-similar tilings
,” Ergodic Theory Dyn. Syst.
17
, 695
–738
(1997
);11.
R. V.
Moody
and N.
Strungaru
, “Point sets and dynamical systems in the autocorrelation topology
,” Can. Math. Bull.
47
(1
), 82
–99
(2004
); arXiv:0705.1142.12.
M.
Baake
, D.
Lenz
, and R. V.
Moody
, “Characterization of model sets by dynamical systems
,” Ergodic Theory Dyn. Syst.
27
, 341
–382
(2007
); arXiv:math/0511648v2.© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
