Rank 1 transformations serve as a source of examples in ergodic theory showing variety algebraic, asymptotic, and spectral properties dynamical systems. The rank one transformation are closely related to the structure semigroup weak limits its powers. In this vein, known new constructions studied.

Let Tl be a transformation on the interval [−1, 1] defined by Chebyshev polynomial of degree l (l ≥ 2), i.e., Tl(cos θ) = cos(lθ). In this paper, we consider Tl as a measure preserving transformation on [−1, 1] with an invariant measure 1 π √ 1−x2 dx. We show that If f(x) is a nonconstant step function with finite kdiscontinuity points with k < l − 1, then it satisfies the Central Limit Theorem...

We generalize the respective “double recurrence” results of Bourgain and of the second author, which established for pairs of L∞ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible measure-preserving point transformations. Our generalizations are set in the context of arbitrary sig...

in this work are studied the jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.

We apply the (C, F )-construction from [Da] to produce a number of funny rank one infinite measure preserving actions of Abelian groups G with “unusual” multiple recurrence properties. In particular, we construct the following for each p ∈ N ∪ {∞}: (i) a p-recurrent action T = (Tg)g∈G such that (if p 6=∞) no one transformation Tg is (p + 1)-recurrent for every element g of infinite order, (ii) ...

This gives us a new probability measure on (Ω,F), so we may define expectations with respect to this conditioned probability measure. Thus for F measurable Y : Ω → R we define the conditional expectation E[Y | X = x] by taking the expectation of Y with respect to the measure (1.1). Consider now how to generalize the idea of conditional probability to the case when P (X = x) = 0. We wish to do t...

The recent introduced dressed coordinates are studied in the path-integral approach. These coordinates are defined in the context of a harmonic oscillator linearly coupled to massless scalar field and, it is shown that in this model the dressed coordinates appear as a coordinate transformation preserving the path-integral functional measure. The analysis also generalizes the sum rules establish...