A note on rotations and interval exchange transformations on 3-intervals

We prove the conjecture that an interval exchange transformation on 3-intervals with corresponding permutation (1; 2; 3) ! (3; 2; 1); and rationally independent discontinuity points, is never measure theoretically isomorphic to an irrational rotation

On the approximation by Lüroth Series

Let x 2 (0; 1] and pn=qn; n 1 be its sequence of Luroth Series convergents. Dene the approximation coecients n = n(x) by n = qnx. In [BBDK] the limiting distribution of the sequence (n)n1 was obtained for a.e. x using the natural extension of the ergodic system underlying the L? uroth Series expansion. Here we show that this can be done without the natural extension. We also will get a bound on the speed of convergence. Using the natural extension we will study the distribution for a.e. x of the sequence (n; n+1)n1 and related sequences like (n + n+1)n1. It turns out that for a.e. x the sequence (n; n+1)n1 is distributed according to a continuous singular distribution function G. Furthermore we will see that two consecutive 's are positively correlated

Random entropy and recurrence

We show that a cocycle, which is nothing but a generalized random walk with index set ℤd, with bounded step sizes is recurrent whenever its associated random entropy is zero, and transient whenever its associated random entropy is positive. This generalizes a well-known one-dimensional result and implies a Polya type dichotomy for this situation

The mean ratio set for ax+b valued cocycles

Let X = Q 1 i=1 Z `(i) be acted upon by the group Q i=1 `(i) i=1 `(i) of changes in nitely many coordinates and a G-measure on X which is nonsingular for the ax+b group. We give a structure theorem for such cocycles, we dene the mean ratio set which is a closed subgroup of the ax + b group and we exhibit for each closed subgroup a cocycle whose mean ratio set is the given subgroup

'The mother of all continued fractions'

In this paper we give the relationship between the regular continued fraction and the Lehner fractions using a procedure known as insertion Starting from the regular continued fraction expansion of any real irrational x when the maximal number of insertions is applied one obtains the Lehner fraction of x Insertions and singularizations show how these and other continued fractions expansions are related We will also investigate the relation between the Lehner fractions and the Farey expansion and obtain the ergodic system underlying the Farey expansio

Домашня та шкільна освіта старообрядців у ХVІІІ – на початку ХХ ст.: аспекти історії

The random β-transformation K is isomorphic to a full shift. This relation gives an invariant measure for K that yields the Bernoulli convolution by projection. We study the local dimension of the invariant measure for K for special values of β and use the projection to obtain results on the local dimension of the Bernoulli convolution

A Gauss-Kusmin theorem for optimal continued fractions

One of the first and still one of the most important results in the metrical theory of continued fractions is the so-called Gauss-Kusmin theorem. Let and let be the regular continued fraction (RCF) expansion of then it was observed by Gauss in 1800 that

Entropy for random group actions

We consider the entropy of systems of random transformations, where the transformations are chosen from a set of generators of a Z d action. We show that the classical denition gives unsatisfactory entropy results in the higher-dimensional case, i.e. when d 2. We propose a denition of the entropy for random group actions which agrees with the classical denition in the one-dimensional case, and which gives satisfactory results in higher dimensions. This denition is based on the bre entropy of a certain skew product. We identify the entropy by an explicit formula which makes it possible to compute the entropy in certain cases

Combinatorics of linear iterated function systems with overlaps

Let $\bm p_0,...,\bm p_{m-1}$ be points in ${\mathbb R}^d$, and let $\{f_j\}_{j=0}^{m-1}$ be a one-parameter family of similitudes of ${\mathbb R}^d$: $$f_j(\bm x) = \lambda\bm x + (1-\lambda)\bm p_j, j=0,...,m-1,$$ where $\lambda\in(0,1)$ is our parameter. Then, as is well known, there exists a unique self-similar attractor $S_\lambda$ satisfying $S_\lambda=\bigcup_{j=0}^{m-1} f_j(S_\lambda)$. Each $\bm x\in S_\lambda$ has at least one address $(i_1,i_2,...)\in\prod_1^\infty\{0,1,...,m-1\}$, i.e., $\lim_n f_{i_1}f_{i_2}... f_{i_n}({\bf 0})=\bm x$. We show that for $\lambda$ sufficiently close to 1, each $\bm x\in S_\lambda\setminus\{\bm p_0,...,\bm p_{m-1}\}$ has $2^{\aleph_0}$ different addresses. If $\lambda$ is not too close to 1, then we can still have an overlap, but there exist $\bm x$'s which have a unique address. However, we prove that almost every $\bm x\in S_\lambda$ has $2^{\aleph_0}$ addresses, provided $S_\lambda$ contains no holes and at least one proper overlap. We apply these results to the case of expansions with deleted digits. Furthermore, we give sharp sufficient conditions for the Open Set Condition to fail and for the attractor to have no holes. These results are generalisations of the corresponding one-dimensional results, however most proofs are different.Comment: Accepted for publication in Nonlinearit

The natural extension of the β-transformation

For each real number β>1 the β-transformation is dened by Tβx = βx(mod1). In this paper the natural extension Tβ of the ergodic system underlying Tβ is explicitly given. Furthermore, it is shown that a certain induced system of this natural extension is Bernoulli. Since Tβ is weakly mixing, due to W. Parry, it follows from a deep result of A. Saleski that the natural extension is also Bernoulli, a result previously obtained by M. Smorodinsky