乘2与乘3
晨光透过迷雾，眼前显现Furstenberg描绘的奇幻动力世界：
猜想A（1969）：考虑[0,1]上的映射T_m(x)=mx (mod1)，m=2、3. 记O(m,x)为x 在T_m映射下的轨道闭包，dim为集合的Hausdorff维数。则对[0,1]上的任意无理数x,
dim O(2,x)+ dim O(3,x) >= 1.
猜想B（1969）：设A，B 分别为[0,1]上的T_2 和T_3 不变闭集，C 为A 与B 的交集。则
dim C <= max{0, dim A + dim B -1}.
3 心2 意，1 探何求？貌合神离，维数怎晓？12 月22 日1114 教室，吴猛博士将为您一一解析其中原委、讲述他对猜想B 的完整证明和对猜想A 除去一个零维集成立的论证。
On the multiplications by 2 and by 3
In one of Furstenberg’s paper in 1969, he proposed two fascinating conjectures:
Conjecture A: Consider the map T_m on [0,1], where T_m(x)=mx (mod 1), m=2 or 3. Let O(m, x) be the closure of the orbit of x under T_m and let dim denote the Hausdorff dimension of a set. Then for every irrational x of [0,1],
dim O(2,x)+ dim O(3,x) >= 1.
Conjecture B: Let A and B be T_2 and T_3 invariant closed subsets of [0,1], respectively, and let C be the intersection of A and B. Then
dim C <= max{0, dim A + dim B -1}.
What are the meanings of these two conjectures and why Furstenberg thought they are true? On Dec. 22, in room 1114, Dr. WU, Meng will explain this in details, give his complete solution to Conjecture B using Furstenberg’s deep insights and confirm Conjecture A for x outside a 0-dimensional set.
