We establish existence and uniqueness of a canonical form for isometric extensions of an ergodic non-singular transformation T. This is applied to describe the structure of commutors of the isometric extensions. Moreover, for a compact group G, we construct a G-valued T-cocycle α which generates the ergodic skew product extension and admits a prescribed subgroup in the centralizer of .
We develop a relative isomorphism theory for random Bernoulli shifts by showing that any random Bernoulli shifts are relatively isomorphic if and only if they have the same fibre entropy. This allows the identification of random Bernoulli shifts with standard Bernoulli shifts.
We present a general theorem describing the isomorphisms of the local Lie algebra structures on the spaces of smooth (real-analytic or holomorphic) functions on smooth (resp. real-analytic, Stein) manifolds, as, for example, those given by Poisson or contact structures. We admit degenerate structures as well, which seems to be new in the literature.
We give a full description of the semiclassical spectral theory of quantum toric integrable systems using microlocal analysis for Toeplitz operators. This allows us to settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of the system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and commuting Hamiltonians. This type of...
Let be the set of all rational maps of degree d ≥ 2 on the Riemann sphere, expanding on their Julia set. We prove that if and all, or all but one, critical points (or values) are in the basin of immediate attraction to an attracting fixed point then there exists a polynomial in the component H(f) of containing f. If all critical points are in the basin of immediate attraction to an attracting fixed point or a parabolic fixed point then f restricted to the Julia set is conjugate to the shift...
We describe totally dissipative parabolic extensions of the one-sided Bernoulli shift. For the fractional linear case we obtain conservative and totally dissipative families of extensions. Here, the property of conservativity seems to be extremely unstable.
The Ito equation is shown to be a geodesic flow of metric on the semidirect product space , where is the group of orientation preserving Sobolev diffeomorphisms of the circle. We also study a geodesic flow of a metric.