Vol 297, No 1 (2017)

Let A ∈ M_n(ℤ) be a matrix with eigenvalues greater than 1 in absolute value. The ℤⁿ-valued random variables ξ_t, t ∈ ℤ, are i.i.d., and P(ξ_t = j) = p_j, j ∈ ℤⁿ, 0 < p₀ < 1, ∑_jp_j = 1. We study the properties of the distributions of the ℝⁿ-valued random variable ζ₁ = ∑_t=1^∞A^−tξ_t and of the random variable ζ = ∑_t=0^∞A^tξ_−t taking integer A-adic values. We obtain a necessary and sufficient condition for the absolute continuity of these distributions. We define an invariant Erdős measure on the compact abelian group of A-adic integers. We also define an A-invariant Erdős measure on the n-dimensional torus. We show the connection between these invariant measures and functions of countable stationary Markov chains. In the case when |{j: p_j ≠ 0}| < ∞, we establish the relation between these invariant measures and finite stationary Markov chains.

We study a family of double confluent Heun equations of the form LE = 0, where L = L_λ,μ,n is a family of second-order differential operators acting on germs of holomorphic functions of one complex variable. They depend on complex parameters λ, μ, and n. The restriction of the family to real parameters satisfying the inequality λ + μ² > 0 is a linearization of the family of nonlinear equations on the two-torus that model the Josephson effect in superconductivity. We show that for all b, n ∈ ℂ satisfying a certain “non-resonance condition” and for all parameter values λ, μ ∈ ℂ, μ ≠ 0, there exists an entire function f_±: ℂ → ℂ (unique up to a constant factor) such that z^−bL(z^bf_±(z^±1)) = d_0± + d_1±z for some d_0±, d_1± ∈ ℂ. The constants d_j,± are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those values λ, μ, n, and b for which the monodromy operator of the corresponding Heun equation has eigenvalue e^2πib. It also gives the description of those values λ, μ, and n for which the monodromy is parabolic, i.e., has a multiple eigenvalue. We consider the rotation number ρ of the dynamical system on the two-torus as a function of parameters restricted to a surface λ + μ² = const. The phase-lock areas are its level sets with nonempty interior. For general families of dynamical systems, the problem of describing the boundaries of the phase-lock areas is known to be very complicated. In the present paper we include the results in this direction that were obtained by methods of complex variables. In our case the phase-lock areas exist only for integer rotation numbers (quantization effect), and their complement is an open set. On their complement the rotation number function is an analytic submersion that induces its fibration by analytic curves. The above-mentioned result on parabolic monodromy implies the explicit description of the union of boundaries of the phase-lock areas as solutions of an explicit transcendental functional equation. For every θ ∉ ℤ we get a description of the set {ρ ≡ ±θ (mod 2ℤ)}.

We review some ergodic and topological aspects of robustly transitive partially hyperbolic diffeomorphisms with one-dimensional center direction. We also discuss step skew-product maps whose fiber maps are defined on the circle which model such dynamics. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents as well as intermingled horseshoes having different types of hyperbolicity. We discuss some recent advances concerning the topology of the space of invariant measures and properties of the spectrum of Lyapunov exponents.

We study combinatorial properties of the subshift induced by the substitution that describes Lysenok’s presentation of Grigorchuk’s group of intermediate growth by generators and relators. This subshift has recently appeared in two different contexts: on the one hand, it allowed embedding Grigorchuk’s group in a topological full group, and on the other hand, it was useful in the spectral theory of Laplacians on the associated Schreier graphs.

It is shown that if a closed smooth orientable manifold Mⁿ, n ≥ 3, admits a Morse–Smale system without heteroclinic intersections (the absence of periodic trajectories is additionally required in the case of a Morse–Smale flow), then this manifold is homeomorphic to the connected sum of manifolds whose structure is interconnected with the type and number of points that belong to the non-wandering set of the Morse–Smale system.

For any α ∈ (0, 1), c ∈ ℝ₊ \ {1} and γ > 0 and for Lebesgue almost all irrational ρ ∈ (0, 1), any two C^2+α-smooth circle diffeomorphisms with a break, with the same rotation number ρ and the same size of the breaks c, are conjugate to each other via a C¹-smooth conjugacy whose derivative is uniformly continuous with modulus of continuity ω(x) = A|log x|^−γ for some A > 0.

We study the statistical and Milnor attractors of step skew products over the Bernoulli shift. In the case when the fiber is a circle, we prove that for a topologically generic step skew product the statistical and Milnor attractors coincide and are Lyapunov stable. To this end we study some properties of the projection of the attractor onto the fiber, which might be of independent interest. In the case when the fiber is a segment, we give a description of the Milnor attractor as the closure of the union of graphs of finitely many almost everywhere defined functions from the base of the skew product to the fiber.

As applied to the problem of asymptotic integration of linear systems of ordinary differential equations, we propose a reduction of order method that allows one to effectively construct solutions indistinguishable in the growth/decrease rate at infinity. In the case of a third-order equation, we use the developed approach to answer Bellman’s problem on splitting WKB asymptotics of subdominant solutions that decrease at the same rate. For a family of Wigner–von Neumann type potentials, the method allows one to formulate a selection rule for nonresonance values of the parameters (for which the corresponding second-order equation has a Jost solution).

We study an analog of the classical Riemann–Hilbert problem stated for the classes of difference and q-difference systems. We generalize Birkhoff’s existence theorem.