Univoque Bases Of Real Numbers In Expansions In Non-integer Bases

In this thesis,we focus on univoque bases of real numbers in expansions in non-integer bases.Given a positive integer M?1 and a base 1<q?M+1,if the sequence?=?₁?₂···?{0,1,···M}^N*satisfies???,then we say that?is a q-expansion of x.Let U?x?denote the set of bases on which x has a unique expansion,and let U?x?denote the set of corresponding unique expansions.First,we obtain an equality about the Haudorff dimension between the set U?x?and the set U_q,and it follows that the function x???dim_HU?x?is a non-increasing Devil's staircase.Next,we extend the existing results about the Hausdorff dimension of the set U?x?,and determine the critical value that when x passes the set U?x?changes from a positive Hausdorff dimension set to a countable set.Finally,we investigate the topological structure of U?x?,and obtain that when M=1 the set U?x?has isolated points for x>1.