Cesar E. Silva

Rank-one weak mixing for nonsingular transformations

We construct rank-one infinite measure preserving transformations satisfying each of the following dynamical properties: (1) ContinuousL∞ spectrum, conservativek-fold cartesian products but nonergodic cartesian square; (2) ergodick-fold cartesian products; (3) nonconservative cartesian square. We show how to modify the construction of (1) to obtain type IIIλ transformations with similar properties.

Measurable Dynamics of Simple p-adic Polynomials

The p-adic numbers have many fascinating properties that are different from those of the real numbers. These properties are a consequence of the fact that the distance in the p-adics is measured using a non-Archimedian absolute value or norm. In this article we study the dynamics of algebraically defined transformations on the p-adics and see that there is a strong connection between the topological property of minimality, which is easy to check for such transformations, and the measure-theoretic property of ergodicity. We start with a section that introduces the p-adic numbers, including their topology and the relevant measures on them, and then define the basic notions from dynamics that we require. In section 4 we show that minimal isometries on subsets of the p-adics are defined on finite unions of balls and are never totally ergodic. In Theorem 4.4 we give a new short proof that for an invertible isometry of a compact open subset of the p-adics minimality implies unique ergodicity and that ergodicity implies minimality. (As we discuss, this theorem is known in a more general context.) Thus for isometries on compact open subsets of the p-adics, the properties of minimality, ergodicity, and unique ergodicity are equivalent. In later sections we study classes of transformations defined by multiplication, translation, and monomial mappings. We demonstrate that they are minimal (hence uniquely ergodic) on balls or, in some examples, spheres in Qp. Many of these results are known, but we present a unified and different treatment of them. Dynamics on the p-adics has been the focus of several researchers recently. While most of this work deals with topological or complex dynamical properties of the p-adics, for which the reader may refer to [1], [4], [13], [17], and the references cited in these articles, measurable dynamics on the p-adics has also received some exposure, particularly in [14], [2], [3] [12], [7], and [8].

Rank-one power weakly mixing non-singular transformations

We show that Chacons non-singular type III _\lambda transformation T_\lambda , 0 , is power weakly mixing, i.e. for all sequences of non-zero integers \{k_{1},\dotsc,k_{r}\} , T_\lambda^{k_{1}}\times\dotsb\times T_\lambda^{k_{r}} is ergodic. We then show that in infinite measure, this condition is not implied by infinite ergodic index (having all finite Cartesian products ergodic), and that infinite ergodic index does not imply 2-recurrence.

Onμ-recurrent nonsingular endomorphisms

We show that the Maharam skew product ofμ-recurrent nonsingular endomorphisms is conservative and give some applications. Among them is the construction of a conservative ergodic invertible natural extension forμ-recurrent ergodic nonsingular endomorphisms.

Multiple and Polynomial Recurrence for Abelian Actions in Infinite Measure

The

On measure-preserving mathcal C1 transformations of compact-open subsets of non-archimedean local fields

(C,F)

The Subadditive Ergodic Theorem and Recurrence Properties of Markovian Transformations

-construction from a previous paper of the first author is applied to produce a number of funny rank one infinite measure preserving actions of discrete countable Abelian groups

ON RATIONALLY ERGODIC AND RATIONALLY WEAKLY MIXING RANK-ONE TRANSFORMATIONS

G

Noninvertible transformations admitting no absolutely continuous -finite invariant measure

with ‘unusual’ multiple recurrence properties. In particular, the following are constructed for each

Digraph representations of rational functions over the p-adic numbers

p\in\Bbb N\cup\{\infty\}