Nikita Sidorov | The University of Manchester (original) (raw)
Papers by Nikita Sidorov
LMS Journal of Computation and Mathematics, 2010
Let β ∈ (1, 2) be a Pisot number and let H β denote Garsia's entropy for the Bernoulli convolutio... more Let β ∈ (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 ≥ 2) Garsia's entropy has been evaluated with high precision by Alexander and Zagier for m = 2 and later by Grabner Kirschenhofer and Tichy for m ≥ 3, and it proves to be close to 1. No other numerical values for H β are known.
Ergodic Theory and Dynamical Systems, 2013
Let 0 < a < b < 1 and let T be the doubling map. Set J (a, b) := {x ∈ [0, 1] : T n x / ∈ (a, b), ... more Let 0 < a < b < 1 and let T be the doubling map. Set J (a, b) := {x ∈ [0, 1] : T n x / ∈ (a, b), n ≥ 0}. In this paper we completely characterize the holes (a, b) for which any of the following scenarios holds:
Abstract: The joint spectral radius of a bounded set of dxd real matrices is defined to be the ma... more Abstract: The joint spectral radius of a bounded set of dxd 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 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.
Abstract. For a map S: X→ X and an open connected set (= a hole) H⊂ X we define JH (S) to be the ... more Abstract. For a map S: X→ X and an open connected set (= a hole) H⊂ X we define JH (S) to be the set of points in X whose S-orbit avoids H. We say that a hole H0 is supercritical if (i) for any hole H such that H0⊂ H the set JH (S) is either empty or contains only fixed points of S;(ii) for any hole H such that H⊂ H0 the Hausdorff dimension of JH (S) is positive.
Let T be an algebraic automorphism of Tm having the following property: the characteristic polyno... more Let T be an algebraic automorphism of Tm having the following property: the characteristic polynomial of its matrix is irreducible over Q, and a Pisot number ß is one of its roots. We define the mapping? t acting from the two-sided ß-compactum onto Tm as follows: ϕ t (̄ ε)= ∑ k ∈ Z ε _k T^-kt, where t is a fundamental homoclinic point for T, ie, a point homoclinic to 0 such that the linear span of its orbit is the whole homoclinic group (provided that such a point exists). We call such a mapping an arithmetic coding of T.
We consider special maps from a symbolic compactum to a two-dimensional torus that connect a bidi... more We consider special maps from a symbolic compactum to a two-dimensional torus that connect a bidirectional translation with a given hyperbolic automorphism of a two-dimensional torus and preserve the maximum entropy measure. In contrast to the known geometric coding method with the aid of Markovian partitions (see references in [2] and [9]), the coding under consideration uses the arithmetic and group theoretic structure.
In [20] the author and A. Vershik have shown that for ß= 1/2 (1+ v5) and the alphabet {0, 1} the ... more In [20] the author and A. Vershik have shown that for ß= 1/2 (1+ v5) and the alphabet {0, 1} the infinite Bernoulli convolution (= the Erdos measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type II 1 under the ß-shift, and the natural extension of the ß-shift provided with the measure equivalent to the Erdos measure, is Bernoulli. In this note we extend this result to all Pisot parameters ß (modulo some general arithmetic conjecture) and an arbitrary “sufficient” alphabet.
Monatshefte für Mathematik, Jan 1, 1998
We de ne a two-sided analog of Erd os measure on the space of two-sided expansions with respect t... more We de ne a two-sided analog of Erd os measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erd os measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erd os and Lebesgue measures on T 2 which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erd os measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.
American Mathematical Monthly, Jan 1, 2003
Journal of dynamical and control systems, Jan 1, 1998
We study the arithmetic codings of hyperbolic automorphisms of the 2torus, i.e. the continuous ma... more We study the arithmetic codings of hyperbolic automorphisms of the 2torus, i.e. the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of 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 T 2 . 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 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.
Mathematical Research Letters, Jan 1, 2001
Topics in dynamics and ergodic theory, Jan 1, 2003
This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call... more 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:
Journal of Theoretical Probability, Jan 1, 2002
. . , f m } is a set of Lipschitz maps of R d . We form the iterated function system (IFS) by ind... more . . , f m } is a set of Lipschitz maps of R d . We form the iterated function system (IFS) by independently choosing the maps so that the map f i is chosen with probability p i ( m i=1 p i = 1). We assume that the IFS contracts on average. We give an upper bound for the Hausdorff dimension of the invariant measure induced on R d and as a corollary show that the measure will be singular if the modulus of the entropy i p i log p i is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's 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.
Nonlinearity, Jan 1, 2004
We consider the iterated function systems (IFSs) that consist of three general similitudes in the... more 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 λ ∈ (0, 1).
Monatshefte für Mathematik, Jan 1, 2005
We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to... more We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the β-numeration. A matrix decomposition of these measures is obtained in the case when β is a PV number. We also determine their Gibbs properties for β being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.
Acta Arith, Jan 1, 1999
We first present a general statement on the number of representations related to a linear recurre... more We first present a general statement on the number of representations related to a linear recurrent basis. Let r be an integer, r ≥ 1, and let a 1 , . . . , a r be reals. We consider a sequence (G k ) k≥0 such that for any k ≥ r,
Arxiv preprint math/9912194, Jan 1, 1999
To a given Pisot unit β we associate a finite abelian group whose size appears to be equal to the... more To a given Pisot unit β we associate a finite abelian group whose size appears to be equal to the discriminant of β. We call it the Pisot group and find its representation in the two-sided β-compactum in the case of β satisfying the relation F in(β) = Z[β] ∩ [0, 1). As a motivation for the definition, we show that the Pisot group is the kernel of some important arithmetic coding of the toral automorphism given by the companion matrix naturally associated with β.
Nonlinearity, Jan 1, 2007
Let p 0 , . . . , p m−1 be points in R d , and let {f j } m−1 j=0 be a one-parameter family of si... more Let p 0 , . . . , p m−1 be points in R d , and let {f j } m−1 j=0 be a one-parameter family of similitudes of R d :
Advances in Mathematics, Jan 1, 2011
The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum pos... more The joint spectral radius of a finite set of real d × 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 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 × 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 × 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
Periodica Mathematica Hungarica, Jan 1, 2003
Given β ∈ (1, 2), a β-expansion of a real x is a power series in base β with coefficients 0 and 1... more Given β ∈ (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 ∈ (1, 2) and a.e. x ∈ (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.
LMS Journal of Computation and Mathematics, 2010
Let β ∈ (1, 2) be a Pisot number and let H β denote Garsia's entropy for the Bernoulli convolutio... more Let β ∈ (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 ≥ 2) Garsia's entropy has been evaluated with high precision by Alexander and Zagier for m = 2 and later by Grabner Kirschenhofer and Tichy for m ≥ 3, and it proves to be close to 1. No other numerical values for H β are known.
Ergodic Theory and Dynamical Systems, 2013
Let 0 < a < b < 1 and let T be the doubling map. Set J (a, b) := {x ∈ [0, 1] : T n x / ∈ (a, b), ... more Let 0 < a < b < 1 and let T be the doubling map. Set J (a, b) := {x ∈ [0, 1] : T n x / ∈ (a, b), n ≥ 0}. In this paper we completely characterize the holes (a, b) for which any of the following scenarios holds:
Abstract: The joint spectral radius of a bounded set of dxd real matrices is defined to be the ma... more Abstract: The joint spectral radius of a bounded set of dxd 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 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.
Abstract. For a map S: X→ X and an open connected set (= a hole) H⊂ X we define JH (S) to be the ... more Abstract. For a map S: X→ X and an open connected set (= a hole) H⊂ X we define JH (S) to be the set of points in X whose S-orbit avoids H. We say that a hole H0 is supercritical if (i) for any hole H such that H0⊂ H the set JH (S) is either empty or contains only fixed points of S;(ii) for any hole H such that H⊂ H0 the Hausdorff dimension of JH (S) is positive.
Let T be an algebraic automorphism of Tm having the following property: the characteristic polyno... more Let T be an algebraic automorphism of Tm having the following property: the characteristic polynomial of its matrix is irreducible over Q, and a Pisot number ß is one of its roots. We define the mapping? t acting from the two-sided ß-compactum onto Tm as follows: ϕ t (̄ ε)= ∑ k ∈ Z ε _k T^-kt, where t is a fundamental homoclinic point for T, ie, a point homoclinic to 0 such that the linear span of its orbit is the whole homoclinic group (provided that such a point exists). We call such a mapping an arithmetic coding of T.
We consider special maps from a symbolic compactum to a two-dimensional torus that connect a bidi... more We consider special maps from a symbolic compactum to a two-dimensional torus that connect a bidirectional translation with a given hyperbolic automorphism of a two-dimensional torus and preserve the maximum entropy measure. In contrast to the known geometric coding method with the aid of Markovian partitions (see references in [2] and [9]), the coding under consideration uses the arithmetic and group theoretic structure.
In [20] the author and A. Vershik have shown that for ß= 1/2 (1+ v5) and the alphabet {0, 1} the ... more In [20] the author and A. Vershik have shown that for ß= 1/2 (1+ v5) and the alphabet {0, 1} the infinite Bernoulli convolution (= the Erdos measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type II 1 under the ß-shift, and the natural extension of the ß-shift provided with the measure equivalent to the Erdos measure, is Bernoulli. In this note we extend this result to all Pisot parameters ß (modulo some general arithmetic conjecture) and an arbitrary “sufficient” alphabet.
Monatshefte für Mathematik, Jan 1, 1998
We de ne a two-sided analog of Erd os measure on the space of two-sided expansions with respect t... more We de ne a two-sided analog of Erd os measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erd os measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erd os and Lebesgue measures on T 2 which is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erd os measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.
American Mathematical Monthly, Jan 1, 2003
Journal of dynamical and control systems, Jan 1, 1998
We study the arithmetic codings of hyperbolic automorphisms of the 2torus, i.e. the continuous ma... more We study the arithmetic codings of hyperbolic automorphisms of the 2torus, i.e. the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of 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 T 2 . 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 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.
Mathematical Research Letters, Jan 1, 2001
Topics in dynamics and ergodic theory, Jan 1, 2003
This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call... more 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:
Journal of Theoretical Probability, Jan 1, 2002
. . , f m } is a set of Lipschitz maps of R d . We form the iterated function system (IFS) by ind... more . . , f m } is a set of Lipschitz maps of R d . We form the iterated function system (IFS) by independently choosing the maps so that the map f i is chosen with probability p i ( m i=1 p i = 1). We assume that the IFS contracts on average. We give an upper bound for the Hausdorff dimension of the invariant measure induced on R d and as a corollary show that the measure will be singular if the modulus of the entropy i p i log p i is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's 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.
Nonlinearity, Jan 1, 2004
We consider the iterated function systems (IFSs) that consist of three general similitudes in the... more 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 λ ∈ (0, 1).
Monatshefte für Mathematik, Jan 1, 2005
We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to... more We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the β-numeration. A matrix decomposition of these measures is obtained in the case when β is a PV number. We also determine their Gibbs properties for β being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.
Acta Arith, Jan 1, 1999
We first present a general statement on the number of representations related to a linear recurre... more We first present a general statement on the number of representations related to a linear recurrent basis. Let r be an integer, r ≥ 1, and let a 1 , . . . , a r be reals. We consider a sequence (G k ) k≥0 such that for any k ≥ r,
Arxiv preprint math/9912194, Jan 1, 1999
To a given Pisot unit β we associate a finite abelian group whose size appears to be equal to the... more To a given Pisot unit β we associate a finite abelian group whose size appears to be equal to the discriminant of β. We call it the Pisot group and find its representation in the two-sided β-compactum in the case of β satisfying the relation F in(β) = Z[β] ∩ [0, 1). As a motivation for the definition, we show that the Pisot group is the kernel of some important arithmetic coding of the toral automorphism given by the companion matrix naturally associated with β.
Nonlinearity, Jan 1, 2007
Let p 0 , . . . , p m−1 be points in R d , and let {f j } m−1 j=0 be a one-parameter family of si... more Let p 0 , . . . , p m−1 be points in R d , and let {f j } m−1 j=0 be a one-parameter family of similitudes of R d :
Advances in Mathematics, Jan 1, 2011
The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum pos... more The joint spectral radius of a finite set of real d × 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 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 × 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 × 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
Periodica Mathematica Hungarica, Jan 1, 2003
Given β ∈ (1, 2), a β-expansion of a real x is a power series in base β with coefficients 0 and 1... more Given β ∈ (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 ∈ (1, 2) and a.e. x ∈ (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.