Related papers: Ergodicity for $p$-adic continued fraction algorit…
Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in S} approach…
We consider a family of $(2,1)$-rational functions given on the set of $p$-adic field $Q_p$. Each such function has a unique fixed point. We study ergodicity properties of the dynamical systems generated by $(2,1)$-rational functions. For…
The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic…
We give a new algorithm of slow continued fraction expansion related to any real cubic number field as a 2-dimensional version of the Farey map. Using our algorithm, we can find the generators of dual substitutions (so-called tiling…
This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of…
A new incremental algorithm for data compression is presented. For a sequence of input symbols algorithm incrementally constructs a p-adic integer number as an output. Decoding process starts with less significant part of a p-adic integer…
A generalization of the regular continued fractions was given by Burger et al. in 2008 [3]. In this paper we give metric properties of this expansion. For the transformation which generates this expansion, its invariant measure and…
We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural…
We introduce a family of piecewise isometries. This family is similar to the ones studied by Hooper and Schwartz. We prove that a renormalization scheme exists inside this family and compute the Hausdorff dimension of the discontinuity set.…
We introduce and study in detail a special class of backward continued fractions that represents a generalization of R\'enyi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we…
Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…
We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial…
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization…
Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let $A_n(x)$ be the…
The purpose of this paper is twofold. On a technical side, we propose an extension of the Hausdorff distance from metric spaces to spaces equipped with asymmetric distance measures. Specifically, we focus on the family of Bregman…
There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove…
We show that for generalized Baker's transformations there is a parameter domain where we have an absolutely continuous ergodic measure and in direct neighborhood there is a parameter domain where not even the variational principle for…
A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman's subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of…