Karma Dajani

From greedy to lazy expansions and their driving dynamics

Abstract In this paper we study the ergodic properties of non-greedy series expansions to non-integer bases β > 1. It is shown that the so-called ‘lazy’ expansion is isomorphic to the ‘greedy’ expansion. Furthermore, a class of expansions to base β > 1, β ∉ ℤ, ‘in between’ the lazy and the greedy expansions are introduced and studies. It is shown that these expansions are isomorphic to expansions of the form Tx = βx + α (mod 1).

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.

Equipartition of interval partitions and an application to number theory

We give wider application and simpler proofs of results describing the rate at which the digits of one number theoretic expansion determine those of another The proofs are based on general measuretheoretic covering arguments and not on the dynamics of specic maps

Random \beta-expansions

In this paper, random expansions to non-integer bases \beta >1 are studied. For \beta s satisfying \beta^2=n\beta +k (with 1\leq k\leq n ) and \beta^n=\beta^{n-1}+\dotsb + \beta +1 the ergodic properties of such expansions are described.

Generalization of a theorem of Kusmin

A Gauss-Kusmin theorem for the natural extension of the regular continued fraction expansion is given. A generalization of a theorem by D. E. Knuth is obtained by similar techniques.

Optimal expansions in non-integer bases

For a given positive integer m, let A = {0, 1, . . . , m} and q ∈ (m,m+1). A sequence (ci) = c1c2 . . . consisting of elements in A is called an expansion of x if ∞ i=1 ciq−i = x. It is known that almost every x belonging to the interval [0,m/(q − 1)] has uncountably many expansions. In this paper we study the existence of expansions (di) of x satisfying the inequalities n i=1 diq−i ≥ n i=1 ciq−i , n = 1, 2, . . . , for each expansion (ci) of x.

Mixing properties of (alpha, beta)-expansions

Let 0< α < 1 and β > 1. We show that every x ∈ [0, 1] has an expansion of the form x = ∞ ∑ n=1 hn β ∑n i=1 piαn− ∑n i=1 pi (x) , where hi = hi (x) ∈ {0, α/β}, and pi = pi (x) ∈ {0, 1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values of α, we give the relationship of this expansion with the greedy β-expansion.

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.

Random N-continued fraction expansions

Abstract The N -continued fraction expansion is a generalization of the regular continued fraction expansion, where the digits 1 in the numerators are replaced by the natural number N . Each real number has uncountably many expansions of this form. In this article we focus on the case N = 2 , and we consider a random algorithm that generates all such expansions. This is done by viewing the random system as a dynamical system, and then using tools from ergodic theory to analyse these expansions. In particular, we use a recent Theorem of Inoue (2012) to prove the existence of an invariant measure of product type whose marginal in the second coordinate is absolutely continuous with respect to Lebesgue measure. Also some dynamical properties of the system are shown and the asymptotic behaviour of such expansions is investigated. Furthermore, we show that the theory can be extended to the random 3-continued fraction expansion.

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.