Periodic orbits of discretized rotations

Let μ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in L2(μ). We show that if L2(μ) admits an exponential frame, then μ must be of pure type. We also classify various μ that admits either kind of exponential bases, in particular, the discrete measures and their connection with integer tiles. By using this and convolution, we construct a class of singularly continuous measures that has an exponential Riesz basis but no exponential orthonormal basis. It is the first of such kind of examples. This is joint work with Chun-kit Lai and Ka-Sing Lau. Geodesic distances and intrinsic distances on some fractal sets Masanori Hino, Kyoto University The off-diagonal Gaussian asymptotics of the heat kernel density associated with a strong local Dirichlet form is often described by using the intrinsic distance (or the Carnot–Caratheodory distance; cf. [5, 4] and the references therein). When the underlying space has a Riemannian structure, the geodesic distance is defined as well, and it coincides with the intrinsic distance in good situations. Then, what if the underlying space is a fractal set? In typical examples, the heat kernel asymptotics is sub-Gaussian; accordingly, the intrinsic distance vanishes identically. However, if we take (a sum of) energy measures as the underlying measure, we can define the nontrivial intrinsic distance as well as the geodesic distance, and can pose a problem whether they are identical. For the 2-dimensional standard Sierpinski gasket, the affirmative answer has been obtained ([2, 3]) by using detailed information on the transition density with probabilistic arguments. In this talk, I will discuss this problem in a more general framework and provide some partial answers based on purely analytic arguments.