Abstract not found

this paper, we shall use the following notations: p i for the i-th prime number (p 1 = 2, p 2 = 3, p 3 = 5,. . . ), (x) for the number of primes x, !(n) for the number of distinct prime factors of n, n) for the number of prime factors of n counted with multiplicity. We write (n) = ( 1) n) (this is the Liouville function) and (n) = ( 1) !(n) so that (n) is completely multiplicative and (n) is multiplicative, and let LN = f(1); (2); : : : ; (N)g GN = f(1); (2); : : : ; (N)g: For y 1 let y (n) and y (n) denote the multiplicative functions de ned by y (p ( 1) (= (p +1 for p > y y (p 1 (= (p +1 for p > y; respectively, and write LN (y) = f y (1); y (2); : : : ; y (N)g GN (y) = f y (1); y (2); : : : ; y (N)g: Research partially supported by Hungarian National Foundation for Scienti c Research, Grant No. T017433 MKM fund FKFP-0139/1997 and by French-Hungarian APAPE-OMFB exchange program F5 /97

this paper, based on notes by R. Beals and M. Spivak, methods of nite semigroups were introduced to obtain some of the results of G. Hedlund.

Let b ≥ 2 be an integer. We prove that the b-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.

This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patch-counting function NX(T) of radius T being finite for all T. A Delone set X of finite type is repetitive if there is a function MX(T) such that every closed ball of radius MX(T)+T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with R n-action. There is a lower bound for MX(T) in terms of NX(T), namely MX(T) ≥ c(NX(T)) 1/n for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for MX(T) purely in terms of NX(T). The complexity of a repetitive Delone set of finite type is measured by the growth rate of its repetitivity function MX(T). For example, the function MX(T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if MX(T) = O(T) as T → ∞ and is densely repetitive if MX(T) = O(NX(T)) 1/n as T → ∞. We show that linearly repetitive sets

There are circular square-free words of length n on three symbols for n 18. This proves a conjecture of R. J. Simpson.

A word u is called 1-balanced if for any two factors v and w of u of equal length, we have 1 jvj i jwj i 1 for each letter i, where jvj i denotes the number of occurrences of i in the factor v. The aim of this paper is to extend the notion of balance to multi-dimensional words. We rst characterize all 1-balanced words on Z n . In particular we prove they are fully periodic for n > 1. We then give a quantitative measure of balancedness for some words on Z 2 with irrational density, including twodimensional Sturmian words. 1

The number of factors of length m of Sturmian words is known to be 1 + P m 1 (m \Gamma i + 1)OE(i): We give a geometric proof of this formula, based on duality and on Euler's relation for planar graphs.

We study a particular case of the two-dimensional Steinhaus theorem, giving estimates of the possible distances between points of the form kff and kff + fi on the unit circle, through an approximation algorithm of fi by the points kff. This allows us to compute covering numbers (maximal measure of Rokhlin stacks having some prescribed regularity properties) for the symbolic dynamical systems associated to the rotation of argument ff acting on the partition of the circle by the points 0, fi. We can then compute topological and measuretheoretic covering numbers for exchange of three intervals; in this way, we prove that every ergodic exchange of three intervals has simple spectrum, and build a new class of three-interval exchanges which are not of rank one.

We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.