arXiv subjects

Nikita Sidorov

Publications and source records attributed to Nikita Sidorov.

41 records · Page 3Linked to original sources

arXiv2000

Bijective and general arithmetic codings for Pisot automorphisms of the torus

Let $T$ be an algebraic automorphism of $\mathbb{T}^{m}$ having the following property: the characteristic polynomial of its matrix is irreducible over $\mathbb{Q}$, and a Pisot number $β$ is one of its roots. We define the mapping $ϕ_{\mathbf{t}}$ acting from the two-sided $β$-compactum onto $\mathbb{T}^{m}$ as follows: \[ ϕ_{\mathbf{t}}(\barε)= \sum_{k\in\mathbb{Z}}ε_{k}T^{-k}\mathbf{t}, \] where $\mathbf{t}$ is a fundamental homoclinic point for $T$, i.e., a point homoclinic to $\mathbf{0}$ such that the linear span of its orbit is the whole homoclinic group (provided such a point exists). We call such a mapping an arithmetic coding of $T$. This paper is aimed to show that under some natural hypothesis on $β$ (which is apparently satisfied for all Pisot units) the mapping $ϕ_{\mathbf{t}}$ is bijective a.e. with respect to the Haar measure on the torus. Besides, we study the case of more general parameters $\mathbf{t}$, not necessarily fundamental, and relate the number of preimages of $ϕ_{\mathbf{t}}$ to certain number-theoretic quantities. We also give several full criteria for $T$ to admit a bijective arithmetic coding. This work continues the study begun in [Sidorov & Vershik, Journal Dynam. Control Systems 4 (1998), 365-399] for the special case $m=2$.

math.DS↗

arXiv2000

On the fine structure of stationary measures in systems which contract-on-average

Suppose $\{f_1,...,f_m\}$ is a set of Lipschitz maps of $\mathbb{R}^d$. We form the iterated function system (IFS) by independently choosing the maps so that the map $f_i$ is chosen with probability $p_i$ ($\sum_{i=1}^m p_i=1$). We assume that the IFS contracts on average. We give an upper bound for the Hausdorff dimension of the invariant measure induced on $\mathbb{R}^d$ and as a corollary show that the measure will be singular if the modulus of the entropy $\sum_i p_i \log p_i$ 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 $\mathbb{R}$.

math.PR↗

arXiv1999

An arithmetic group associated with a Pisot unit and its symbolic-dynamical representatiom

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 $Fin(β)=\Bbb Z[β]\cap[0,1)$. 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 $β$.

math.NT↗

arXiv1998

Ergodic properties of Erdös measure, the entropy of the goldenshift, and related problems

We define a two-sided analog of Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on $\Bbb T^2$ 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 Erdös 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.

math.DS↗

arXiv1998

Bijective arithmetic codings of hyperbolic automorphisms of the 2-torus, and binary quadratic forms

We 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 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 the 2-torus. 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.

math.DS↗