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Nikita Sidorov

Publications and source records attributed to Nikita Sidorov.

At least 37 records · Page 2Linked to original sources
arXiv2013

The doubling map with asymmetrical holes

Let $0 0$; $\mathcal J(a,b)$ is uncountable of zero Hausdorff dimension; $\mathcal J(a,b)$ is of positive Hausdorff dimension. {enumerate} In particular, we show that (iv) is always the case if \[ b-a<\frac14\prod_{n=1}^\infty \bigl(1-2^{-2^n}\bigr)\approx 0.175092 \] and that this bound is sharp. As a corollary, we give a full description of first and second order critical holes introduced in \cite{SSC} for the doubling map. Furthermore, we show that our model yields a continuum of "routes to chaos" via arbitrary sequences of products of natural numbers, thus generalizing the standard route to chaos via period doubling.

math.DS↗
arXiv2013

On a Devil's staircase associated to the joint spectral radii of a family of pairs of matrices

The joint spectral radius of a finite set of real d x d matrices is defined to be the maximum possible exponential rate of growth of products of matrices drawn from that set. In previous work with K. G. Hare and J. Theys we showed that for a certain one-parameter family of pairs of matrices, this maximum possible rate of growth is attained along Sturmian sequences with a certain characteristic ratio which depends continuously upon the parameter. In this paper we answer some open questions from that paper by showing that the dependence of the ratio function upon the parameter takes the form of a Devil's staircase. We show in particular that this Devil's staircase attains every rational value strictly between 0 and 1 on some interval, and attains irrational values only in a set of Hausdorff dimension zero. This result generalises to include certain one-parameter families considered by other authors. We also give explicit formulas for the preimages of both rational and irrational numbers under the ratio function, thereby establishing a large family of pairs of matrices for which the joint spectral radius may be calculated exactly.

math.OC↗
arXiv2013

Extremal sequences of polynomial complexity

The joint spectral radius of a bounded set of $d \times d$ real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called \emph{extremal} if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer $p \geq 1$, there exists a pair of square matrices of dimension $2^p(2^{p+1}-1)$ for which every extremal sequence has subword complexity at least $2^{-p^2}n^p$.

math.OC↗
arXiv2011

Optimizing Properties of Balanced Words

In the past few decades there has been a good deal of papers which are concerned with optimization problems in different areas of mathematics (along 0-1 words, finite or infinite) and which yield - sometimes quite unexpectedly - balanced words as optimal. In this note we list some key results along these lines known to date.

cs.DM↗
arXiv2011

On the topology of sums in powers of an algebraic number

Let $1<q<2$ and \[ Λ(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $Λ(q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $Λ(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $α$ such that $q|α|<1$, then $Λ(q)$ is dense in $\mathbb{R}$.

math.NT↗
arXiv2010

An explicit counterexample to the Lagarias-Wang finiteness conjecture

The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the \emph{finiteness property} if there exists a periodic product which achieves this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real $d \times d$ matrices satisfies the finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the finiteness conjecture exist, showing in particular that there exists a family of pairs of $2 \times 2$ matrices which contains a counterexample. Similar results were subsequently given by V.D. Blondel, J. Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample to the finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the set \[ \mathsf{A}_{α_*}:= \{({cc}1&1\\0&1), α_*({cc}1&0\\1&1)\}\] we give an explicit value of α_* \simeq 0.749326546330367557943961948091344672091327370236064317358024...] such that $\mathsf{A}_{α_*}$ does not satisfy the finiteness property.

math.OC↗
arXiv2009

Growth rate for beta-expansions

Let $β>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{β-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where $ε_k\in\{0,1,...,m-1\}$ for all $k$ (a $β$-expansion of $x$). It is known that a.e. $x\in I_β$ has a continuum of distinct $β$-expansions. In this paper we prove that if $β$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $β$. When $β<\frac{1+\sqrt5}2$, we show that the set of $β$-expansions grows exponentially for every internal $x$.

math.NT↗
arXiv2009

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

Let $β\in(1,2)$ be a Pisot number and let $H_β$ denote Garsia's entropy for the Bernoulli convolution associated with $β$. Garsia, in 1963 showed that $H_β<1$ for any Pisot $β$. For the Pisot numbers which satisfy $x^m=x^{m-1}+x^{m-2}+...+x+1$ (with $m\ge2$) Garsia's entropy has been evaluated with high precision by Alexander and Zagier and later improved by Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other numerical values for $H_β$ are known. In the present paper we show that $H_β>0.81$ for all Pisot $β$, and improve this lower bound for certain ranges of $β$. Our method is computational in nature.

math.DS↗
arXiv2009

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

Let $q\in(1,2)$; it is known that each $x\in[0,1/(q-1)]$ has an expansion of the form $x=\sum_{n=1}^\infty a_nq^{-n}$ with $a_n\in\{0,1\}$. It was shown in \cite{EJK} that if $q<(\sqrt5+1)/2$, then each $x\in(0,1/(q-1))$ has a continuum of such expansions; however, if $q>(\sqrt5+1)/2$, then there exist infinitely many $x$ having a unique expansion \cite{GS}. 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 $q 1$ there exists $\ga_m>0$ such that for any $q\in(2-\ga_m,2)$, there exists $x$ which has exactly $m$ expansions in base $q$.

math.NT↗
arXiv2008

On the average growth exponent for beta-expansions

Let $\be\in(1,2)$. Each $x\in I_\be:=[0,\frac{1}{\be-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty a_k\be^{-k}, \] where $a_k\in\{0,1\}$ for all $k$ (a $\be$-expansion of $x$). It was shown in \cite{S} that a.e. $x\in I_\be$ has a continuum of distinct $\be$-expansions. In this paper we show that for a generic $x$, this continuum has one and the same growth rate, i.e., the general $\be$-expansions exhibit an ergodic behaviour. When $\be<\frac{1+\sqrt5}2$, we show that the set of $\be$-expansions grows exponentially for every $x\in(0,\frac{1}{\be-1})$. Special attention is paid to the case $\be=\frac{1+\sqrt5}2$, for which we explicitly compute the average growth exponent and apply this result to evaluating the local dimension of the corresponding Bernoulli convolution at a Lebesgue-generic $x$.

math.DS↗
arXiv2008

Periodic unique beta-expansions: the Sharkovskii ordering

Let $β\in(1,2)$. Each $x\in[0,\frac{1}{β-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where $ε_k\in\{0,1\}$ for all $k$ (a $β$-expansion of $x$). If $β>\frac{1+\sqrt5}{2}$, then, as is well known, there always exist $x\in(0,\frac1{β-1})$ which have a unique $\be$-expansion. In the present paper we study (purely) periodic unique $β$-expansions and show that for each $n\ge2$ there exists $β_n\in[\frac{1+\sqrt5}{2},2)$ such that there are no unique periodic $β$-expansions of smallest period $n$ for $β\leβ_n$ 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 Sharkovski\uı 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.

math.DS↗
arXiv2007

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 ${\mathbb R}^d$: $$ f_j(\bm x) = λ\bm x + (1-λ)\bm p_j, j=0,...,m-1, $$ where $λ\in(0,1)$ is our parameter. Then, as is well known, there exists a unique self-similar attractor $S_λ$ satisfying $S_λ=\bigcup_{j=0}^{m-1} f_j(S_λ)$. Each $\bm x\in S_λ$ 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 $λ$ sufficiently close to 1, each $\bm x\in S_λ\setminus\{\bm p_0,...,\bm p_{m-1}\}$ has $2^{\aleph_0}$ different addresses. If $λ$ 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 $\bm x\in S_λ$ has $2^{\aleph_0}$ addresses, provided $S_λ$ 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.

math.DS↗
arXiv2003

Golden gaskets: variations on the Sierpiński sieve

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor $\la\in(0,1)$. As is well known, for $\la=1/2$ the invariant set, $§_\la$, is a fractal called the Sierpiński sieve, and for $\la<1/2$ it is also a fractal. Our goal is to study $§_\la$ for this IFS for $1/2<\la<2/3$, i.e., when there are "overlaps" in $§_\la$ as well as "holes". In this introductory paper we show that despite the overlaps (i.e., the Open Set Condition breaking down completely), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic $\la$'s (so-called "multinacci numbers"). We evaluate $\dim_H(§_\la)$ for these special values by showing that $§_\la$ is essentially the attractor for an infinite IFS which does satisfy the Open Set Condition. 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 $\la$ we show that if $\la$ is close to 2/3, then $§_\la$ has a nonempty interior and that if $\la<1/\sqrt{3}$ then §_\la$ has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of the model in question.

math.DS↗
arXiv2002

Spectra of Bernoulli convolutions as multipliers in $L^p$ on the circle

It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution $μ_θ$ parameterized by a Pisot number $θ$, is countable. Combined with results of Salem and Sarnak, this proves that for every fixed $θ>1$ the spectrum of the convolution operator $f\mapsto μ_θ*f$ in $L^p(S^1)$ (where $S^1$ is the circle group) is countable and is the same for all $p\in(1,\infty)$, namely, $\bar{\{\hat{μ_θ}(n) : n\in\mathbb{Z}\}}$. Our result answers the question raised by P. Sarnak in \cite{Sar}. We also consider the sets $\bar{\{\hat{μ_θ}(rn) : n\in\mathbb{Z}\}}$ for $r>0$ which correspond to a linear change of variable for the measure. We show that such a set is still countable for all $r\in\Q(θ)$ but uncountable (a non-empty interval) for Lebesgue-a.e. $r>0$.

math.FA↗
arXiv2002

Universal $β$-expansions

Given $β\in(1,2)$, a $β$-expansion of a real $x$ is a power series in base $β$ with coefficients 0 and 1 whose sum equals $x$. The aim of this note is to study certain problems related to the universality and combinatorics of $β$-expansions. Our main result is that for any $β\in(1,2)$ and a.e. $x\in (0,1)$ there always exists a universal $β$-expansion of $x$ in the sense of Erdös and Komornik, i.e., a $β$-expansion whose complexity function is $2^n$. We also study some questions related to the points having less than a full branching continuum of $β$-expansions and also normal $β$-expansions.

math.DS↗
arXiv2002

Ergodic-theoretic properties of certain Bernoulli convolutions

In [17] the author and A. Vershik have shown that for $\be=\frac12(1+\sqrt5)$ and the alphabet $\{0,1\}$ the infinite Bernoulli convolution ($=$ the Erdös measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type $\mathrm{II}_1$ under the $\be$-shift, and the natural extension of the $\be$-shift provided with the measure equivalent to the Erdös measure, is Bernoulli. In this note we extend this result to all Pisot parameters $\be$ (modulo some general arithmetic conjecture) and an arbitrary "sufficient" alphabet.

math.DS↗
arXiv2002

Arithmetic Dynamics

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the circle; (3) Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodic-theoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create "redundant" representations (those whose space of "digits" is a priori larger than necessary) and study their combinatorics.

math.DS↗