Almost every number has a continuum of beta-expansions

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Invariant densities for random ß-expansions (2007, background): “βn ∈ S1 . Now, suppose the converse is true, i.e., suppose that the measures νβ and mp ⊗ � β,p coincide.”
Expansions in non-integer bases: Lower, middle and top orders (2006, background): “n ∈ Z, each of which has ℵ0 expansions in base q (see [17] for a detailed study of the space of expansions for this case). However, if q > G, then although a.e. x ∈ Iq has 2ℵ0 expansions in base q [15], there always exist (at least countably many) reals having a unique expansion – see [5]. Let Uq”
Hausdorff dimension of univoque sets and Devil's staircase (2015, result): “ [29]. One of the striking features of such bases is that generically a number has a continuum of diﬀerent expansions, a situation quite opposite to that of integer bases; see, e.g., [13] and Sidorov [30]. However, surprising unique expansions Date: Version of 2015-03-02-a. 2000 Mathematics Subject ”
Unique expansions of real numbers (2006, background): “m the above results that Uq is closed for almost every q > 1. • Let q > 1 be a non-integer. In [DDV] it has been proved that almost every x ∈ J has a continuum of expansions in base q (see also [Si1]). It follows from the above results that the set Uq has Lebesgue measure zero. Hence, the set ”