Anzai and Furstenberg Transformations on the 2-torus and Topologically Quasi-Discrete Spectrum

摘要

Let φ_0 be an Anzai transformation on the 2-torus T2 defined by a(x,y) _ (e2-'*?x,xy) and 4>f a Furstenberg transformation on T2 defined by f(x,y) = (ellWx,e2ir^wxy) where 0 is an irrational number and/ is a real valued continuous function on the 1-torus T. In the present note we will show that fy has topologically quasi-discrete spectrum if and only if fy is topologically conjugate to cj>n. Furthermore we will show that for any irrational number 6 there is a real valued continuous function / on T such that ty does not have topologically quasi-discrete spectrum but is uniquely ergodic.