Related papers: On Multiple and Polynomial Recurrent extensions of…
We prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+p_i(n)$, with rationally independent $p_i$'s with zero constant term. This is in contrast to the single…
We discuss multiple versions of rational ergodicity and rational weak mixing for "nice" transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.
Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are…
We extend the authors' previous work on Wiener-Wintner double recurrence theorem to the case of polynomials.
We give a short proof of polynomial recurrence with large intersection for additive actions of finite-dimensional vector spaces over countable fields on probability spaces, improving upon the known size and structure of the set of strong…
We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset $E\subset\Bbb N\cup\{\infty\}$ as the set of essential…
Motivated by partition regularity problems of homogeneous quadratic equations, we prove multiple recurrence and convergence results for multiplicative measure preserving actions with iterates given by rational sequences involving…
Extension dimension is characterized in terms of $\omega$-maps. We apply this result to prove that extension dimension is preserved by refinable maps between metrizable spaces. It is also shown that refinable maps preserve some…
In a recent paper, Melbourne and Terhesiu [Operator renewal theory and mixing rates for dynamical systems with infinite measure, Invent. Math. 189 (2012), 61-110] obtained results on mixing and mixing rates for a large class of…
A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. We show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic…
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the…
By using a similar pattern of arguments, we show that in three categories the collection of isomorphisms forms a residual subset of the space of morphisms. We first consider surjective continuous mappings on Cantor spaces. Next, we look at…
In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the…
We show that the following properties are preserved under inverse limits: countable fan-tightness, q+, discrete generation and selective separability. We also present several examples based on inverse limits of countable spaces.
It is well-known that for any non-constant polynomial $P$ with integer coefficients the sequence $(P(n))_{ n\in \mathbb N}$ has the property that there are infinitely many prime numbers dividing at least one term of this sequence.…
We exhibit a change of variables that maintains the Mahler measure of a given polynomial. This method leads to the construction of highly non-trivial polynomials with given Mahler measure and settles some conjectural numerical formulas due…
We show that (as conjectured by Lin and Wang) when a Vassiliev invariant of type $m$ is evaluated on a knot projection having $n$ crossings, the result is bounded by a constant times $n^m$. Thus the well known analogy between Vassiliev…
It is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are preserved under iterated double Ore extensions are invariants of the Poisson…
We present a theorem about irreducibility of a polynomial that is the resultant of two others polynomials. The proof of this fact is based on the field theory. We also consider the converse theorem and some examples.
Each subset $E\subset\Bbb N$ is realized as the set of essential values of the multiplicity function for the Koopman operator of an ergodic conservative infinite measure preserving transformation.