Nikita Sidorov

Expansions in non-integer bases: Lower, middle and top orders

Abstract Let q ∈ ( 1 , 2 ) ; it is known that each x ∈ [ 0 , 1 / ( q − 1 ) ] has an expansion of the form x = ∑ n = 1 ∞ a n q − n with a n ∈ { 0 , 1 } . It was shown in [P. Erdős, I. Joo, V. Komornik, Characterization of the unique expansions 1 = ∑ i = 1 ∞ q − n i and related problems, Bull. Soc. Math. France 118 (1990) 377–390] that if q ( 5 + 1 ) / 2 , then each x ∈ ( 0 , 1 / ( q − 1 ) ) has a continuum of such expansions; however, if q > ( 5 + 1 ) / 2 , then there exist infinitely many x having a unique expansion [P. Glendinning, N. Sidorov, Unique representations of real numbers in non-integer bases, Math. Res. Lett. 8 (2001) 535–543]. In the present paper we begin the study of parameters q for which there exists x having a fixed finite number m > 1 of expansions in base q. In particular, we show that if q q 2 = 1.71 … , then each x has either 1 or infinitely many expansions, i.e., there are no such q in ( ( 5 + 1 ) / 2 , q 2 ) . On the other hand, for each m > 1 there exists γ m > 0 such that for any q ∈ ( 2 − γ m , 2 ) , there exists x which has exactly m expansions in base q.

Bijective Arithmetic Codings of Hyperbolic Automorphisms of the 2-Torus, and Binary Quadratic Forms

AbstractWe study the arithmetic codings of hyperbolic automorphisms of the 2-torus, i.e., the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of the additive group onto the torus which preserve this structure and turn the two-sided shift into a given automorphism of the torus. This group is uniquely defined by an automorphism, and such an arithmetic coding is a homomorphism of that group onto

Periodic unique beta-expansions: the Sharkovskii ordering

\mathbb{B}^2

Golden gaskets: variations on the Sierpinski sieve

. The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtained; it is formulated in terms of a certain binary quadratic form constructed by means of a given automorphism. Furthermore, we describe all bijective arithmetic codings in terms of the Dirichlet group of the corresponding quadratic field. The minimum of that quadratic form over the nonzero elements of the lattice coincides with the minimal possible order of the kernel of a homomorphism described above.

Bijective and General Arithmetic Codings for Pisot Toral Automorphisms

Suppose {f1,...,fm} is a set of Lipschitz maps of ℝd. We form the iterated function system (IFS) by independently choosing the maps so that the map fi is chosen with probability pi (∑mi=1pi=1). We assume that the IFS contracts on average. We give an upper bound for the upper Hausdorff dimension of the invariant measure induced on ℝd and as a corollary show that the measure will be singular if the modulus of the entropy ∑ipi log pi is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannons Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of ℝ.

Universal β-expansions

Let β ∈(1,2). Each x ∈[0,1/( β −1)] can be represented in the form where e k ∈{0,1} for all k (a β -expansion of x ). If , then, as is well known, there always exist x ∈(0,1/( β −1)) which have a unique β -expansion. We study (purely) periodic unique β -expansions and show that for each n ≥2 there exists such that there are no unique periodic β -expansions of smallest period n for β ≤ β n and at least one such expansion for β > β n . Furthermore, we prove that β k β m if and only if k is less than m in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps.

On the topology of sums in powers of an algebraic number

Let p0, ..., pm−1 be points in , and let be a one-parameter family of similitudes of : where λ (0, 1) is our parameter. Then, as is well known, there exists a unique self-similar attractor Sλ satisfying . Each x Sλ has at least one address , i.e. .We show that for λ sufficiently close to 1, each x Sλ {p0, ..., pm−1} has different addresses. If λ is not too close to 1, then we can still have an overlap, but there exist xs which have a unique address. However, we prove that almost every x Sλ has addresses, provided Sλ 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 generalizations of the corresponding one-dimensional results, however most proofs are different.

A lower bound for Garsia's entropy for certain Bernoulli convolutions

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, with a common contraction factor λ in (0, 1). As is well known, for λ = 1/2 the attractor, S_λ, is a fractal called the Sierpinski sieve and for λ < 1/2 it is also a fractal. Our goal is to study S_λ for this IFS for 1/2 < λ < 2/3 , i.e. when there are ‘overlaps’ in S_λ as well as ‘holes’. In this introductory paper we show that despite the overlaps (i.e. the breaking down of the open set condition (OSC)), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic λ (so-called multinacci numbers). We evaluate the ausdorff dimension of S_λ for these special values by showing that S_λ is essentially the attractor for an infinite IFS that does satisfy the OSC. We also show that the set of points in the attractor with a unique ‘address’ is self-similar and compute its dimension. For non-multinacci values of λ we show that if λ is close to 2/3 , then S_λ has a non-empty interior. Finally we discuss higher-dimensional analogues of the model in question.