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 sequenceε=ε₁ε₂···∈{0,1,···M}^N*satisfies（?）,then we say thatεis 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.