Let X be an infinite compact metric space with finite covering dimension. Let α, β : X → X be two minimal homeomorphisms. Suppose that the range of K0-groups of both crossed products are dense in the space of real affine continuous functions. Suppose also that both α and β have countably many extremal invariant measures. We show that α and β are approximately conjugate uniformly in measure if a...

Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable...

We construct a four-parameter family of Markov processes on infinite Gelfand-Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinite-dimensional boundary of the Gelfand-Tsetlin graph or, equivalently, the space of extreme characters of the infinite-dimensional unitary group U(∞). The process has a unique inva...

A 1-2 model configuration is a subgraph of the hexagonal lattice, in which the edgedegree of each vertex is either 1 or 2. We prove that for any translation invariant Gibbs measure of the 1-2 model configurations on the whole-plane hexagonal lattice, almost surely there are no infinite paths. Using a measure-preserving correspondence between 1-2 model configurations on the hexagonal lattice and...

Under weaker condition than that of Riedi & Mandelbrot, the Hausdorff (and Hausdorff-Besicovitch) dimension of infinite self-similar set K ⊂ R which is the invariant compact set of infinite contractive similarities {Sj(x) = ρjRjx + bj}j∈N (0 < ρj < 1, bj ∈ R , Rj orthogonal) satisfying open set condition is obtained. It is proved (under some additional hypotheses) that the β-mean quadratic vari...

The Ornstein-Uhlenbeck semigroup combines Gaussian diffusion with the flow of a linear vector field. In infinite dimensional settings there can be non-Gaussian invariant measures. This gives a context for one version of the renormalization group. The adjoint of the OrnsteinUhlenbeck semigroup with respect to an invariant measure need not be an Ornstein-Uhlenbeck semigroup. This adjoint is the a...

We show that certain billiard tables with non-compact cusps are mixing with respect to the invariant infinite measure, in the sense of Krengel and Sucheston. Moreover, we show that the scaling rate is slower than a certain polynomial rate.

We consider the sequence of the hyperspheres Mn,r i.e. the homogeneous transitive spaces of the Cartan subgroup SDiag(n, R) of the group SL(n, R), n = 1 . . . , and studied the normalized limit of the corresponding sequence of the invariant measures mn on those spaces. In the case of compact groups and homogeneous spaces, as example for classical pairs (SO(n), Sn−1), n = 1 . . . the limit of co...

Weakly chaotic nonlinear maps with marginal fixed points have an infinite invariant measure. Time averages of integrable and nonintegrable observables remain random even in the long time limit. Temporal averages of integrable observables are described by the Aaronson-Darling-Kac theorem. We find the distribution of time averages of nonintegrable observables, for example, the time average positi...