On the boundary of self affine tilings generated by Pisot numbers

Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to \beta -shifts

“…reducible case, the relation (1) between the two spectra is not as simple: pure discrete spectrum for σ implies that for T t but the opposite implication typically fails. The following conjecture has become known as the Pisot conjecture ( [7]).…”

“…[27]) of the tiling flow; see [5] for a comprehensive discussion. The related number-theoretic investigations have been undertaken in a number of works by Akiyama, Frougny, Ito, Rao, Solomyak, Steiner, and Thuswaldner( [1,2,13,16,29]). For a recent survey of the connections between tilings, Pisot arithmetics, and substitutions consult [7].…”

Substitution Delone sets with pure point spectrum are inter-model sets

“…Throughout correspondence with Valerie Berthe, it is noted that the algebraic coincidence is necessary if an exclusive inner point exists, which is conjectured to hold for every Pisot unit substitution in 1-dimension (see [1]). Does every Pisot unit substitution admit algebraic coincidence?…”

Purely periodic 𝛽-expansions with Pisot unit base

“…So {X i } 1≤i≤q forms a self-similar tiling system (cf. [16,6,2,3,4,7,19,11,8]). For detailed discussions of the tiling and dynamical properties of {X i } 1≤i≤q , we refer to [11,8].…”

“…(It is called a Rauzy fractal by [6,7,19] and is called a central tile by [20,3,4].) Denote by e 1 , .…”