Type: Article
Publication Date: 1996-01-01
Citations: 67
DOI: https://doi.org/10.1090/s0002-9947-96-01640-6
The set of Penrose tilings, when provided with a natural compact metric topology, becomes a strictly ergodic dynamical system under the action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by translation. We show that this action is an almost 1:1 extension of a minimal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> action by rotations on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper T Superscript 4"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">T</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {T}^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, i.e., it is an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generalization of a <italic>Sturmian dynamical system</italic>. We also show that the inflation mapping is an almost 1:1 extension of a hyperbolic automorphism on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper T Superscript 4"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">T</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {T}^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The local topological structure of the set of Penrose tilings is described, and some generalizations are discussed.