On a class of Rauzy fractals without the finiteness property
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2019-01-01
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We present some topological and arithmetical aspects of a class of Rauzy fractals a,b related to the polynomials of the form Pa,b (x) = x3 − ax2 − bx − 1, where a and b are integers satisfying −a + 1 ≤ b ≤ −2. This class has the property that 0 lies on the boundary of a,b. We construct explicit finite automata that recognize the boundaries of these fractals. This allows to establish the number of neighbors of a,b in the tiling it generates. Furthermore, we prove that if 2a + 3b + 4 ≤ 0 then a,b is not homeomorphic to a topological disk. We also show that the boundary of the set 3,−2 is generated by two infinite iterated function systems.
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Osaka Journal of Mathematics, v. 56, n. 3, p. 577-599, 2019.
