Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in function of the alphabets.
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Vilmos Komornik, Anna Chiara Lai, Marco Pedicini, Generalized golden ratios of ternary alphabets. J. Eur. Math. Soc. 13 (2011), no. 4, pp. 1113–1146DOI 10.4171/JEMS/277