ar X iv : 0 90 2 . 04 88 v 2 [ m at h . N T ] 1 1 M ay 2 00 9 GROWTH RATE FOR BETA - EXPANSIONS

ar X iv : 0 90 2 . 04 88 v 3 [ m at h . N T ] 1 7 A ug 2 00 9 GROWTH RATE FOR BETA - EXPANSIONS

Let β > 1 and let m > β be an integer. Each x ∈ Iβ := [0, m−1 β−1 ] can be represented in the form x = ∞ ∑ k=1 εkβ , where εk ∈ {0, 1, . . . ,m − 1} for all k (a β-expansion of x). It is known that a.e. x ∈ Iβ has a continuum of distinct β-expansions. In this paper we prove that if β is a Pisot number, then for a.e. x this continuum has one and the same growth rate. We also link this rate to th...

ar X iv : m at h / 04 11 14 0 v 2 [ m at h . N T ] 4 M ay 2 00 5 THE 3 x + 1 SEMIGROUP

The 3x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2k+1 3k+2 : k ≥ 0} together with {2}. This semigroup encodes backwards iteration under the 3x + 1 map, and the 3x + 1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals a b in lowest terms with b 6≡ 0( mod 3), and so contains al...

ar X iv : 0 90 5 . 11 71 v 1 [ m at h . N T ] 8 M ay 2 00 9 Ramification of local fields and Fontaine ’ s property ( P m )

The ramification subgroup of the absolute Galois group of a complete discrete valuation field with perfect residue field is characterized by Fontaine’s property (Pm).

ar X iv : h ep - p h / 04 02 19 0 v 2 2 6 M ay 2 00 4 1 Pentaquarks in the 10 F - and 10 F - plets

ar X iv : 0 80 3 . 44 39 v 1 [ m at h . D S ] 3 1 M ar 2 00 8 PERIODIC UNIQUE BETA - EXPANSIONS : THE SHARKOVSKIĬ ORDERING

Let β ∈ (1, 2). Each x ∈ [0, 1 β−1 ] can be represented in the form