Continuity of invariant measures for families of maps

Let {τ_r} be the family of maps from [0,1] into [0,1] with properties similar to those of τ_r(x) = rx(1 ? x), 0 ≤ r ≤ 4. The limiting behaviour of orbits {τ_r^j(x)}_j^∞ = 1 is a complicated and discontinuous function of the parameter r. The stochastic approximation to the difference equation x_{n + 1} = τ_r(x_n), x_{n + 1} = τ_r(x_n) + W, where W is a fixed random variable independent of r and x_n, is considered. It is shown that this Markov process admits a unique absolutely continuous invariant measure μ_r and, furthermore, that the map r → μ_r is continuous. Such a result is important in applications, since slight changes in the shape of τ_r no longer cause discontinuous consequences in the limiting behaviour of the system.     

Quantum invariant families of matrices in free probability

We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued R-cyclicity condition. This is a surprising contrast with the Aldous-Hoover characterization of jointly exchangeable arrays.     

On common invariant cones for families of matrices

The existence and construction of common invariant cones for families of real matrices is considered. The complete results are obtained for 2×2 matrices (with no additional restrictions) and for families of simultaneously diagonalizable matrices of any size. Families of matrices with a shared dominant eigenvector are considered under some additional conditions.     

Carleson and Vanishing Carleson Measures on Radial Trees

We extend a discrete version of an extension of Carleson’s theorem proved in [5] to a large class of trees that have certain radial properties. We introduce the geometric notion of s-vanishing Carleson measure on such a tree T (with s ≥ 1) and give several characterizations of such measures. Given a measure σ on T and p ≥ 1, let L ^p (σ) denote the space of functions g defined on T such that |g| ^p is integrable with respect to σ and let L ^p (? T) be the space of functions f defined on the boundary of T such that |f| ^p is integrable with respect to the representing measure of the harmonic function 1.We prove the following extension of the discrete version of a classical theorem in the unit disk proved by Power. A finite measure σ on T is an s-vanishing Carleson measure if and only if for any real number p > 1, the Poisson operator P : L ^p (? T) → L ^sp (σ) is compact. Characterizations of weak type for the case p = 1 and in terms of the tree analogue of the extended Poisson kernel are also given. Finally, we show that our results hold for homogeneous trees whose forward probabilities are radial and whose backward probabilities are constant, as well as for semihomogeneous trees.     

On families of invariant lines of a Brouwer homeomorphism

ABSTRACT

We present properties of equivalence classes of the codivergency relation defined for a Brouwer homeomorphism for which there exists a family of invariant pairwise disjoint lines covering the plane. In particular, using the codivergency relation we describe the sets of regular and irregular points for such Brouwer homeomorphisms. Moreover, we show that under this assumption the interior of each equivalence class of this relation is invariant and simply connected.     

Quadratic function fields with invariant class group

Emil Artin studied quadratic extensions of k(x) where k is a prime field of odd characteristic. He showed that there are only finitely many such extensions in which the ideal class group has exponent two and the infinite prime does not decompose. The main result of this paper is: If K is a quadratic imaginary extension of k(x) of genus G, where k is a finite field of order q, in which the infinite prime of k(x) ramifies, and if the ideal class group has exponent 2, then q = 9, 7, 5, 4, 3, or 2 and G ≤ 1, 1, 2, 2, 4, and 8, respectively. The method of Artin's proof gives G ≤ 13, 9, and 9724 for q = 7, 5, and 3, respectively. If the infinite prime is inert in K, both the methods of this paper and Artin's methods give bounds on the genus that are roughly double those in the ramified case.     

The Calabi invariant and the Euler class

We show the following relationship between the Euler class for the group of the orientation preserving diffeomorphisms of the circle and the Calabi invariant for the group of area preserving diffeomorphisms of the disk which are the identity along the boundary. A diffeomorphism of the circle admits an extension which is an area preserving diffeomorphism of the disk. For a homomorphism from the fundamental group of a closed surface to the group of the diffeomorphisms of the circle, by taking the extensions for the generators , one obtains the product of their commutators, and this is an area preserving diffeomorphism of the disk which is the identity along the boundary. Then the Calabi invariant of this area preserving diffeomorphism is a non-zero multiple of the Euler class of the associated circle bundle evaluated on the fundamental cycle of the surface.

     

Specific features of families of invariant manifolds of conservative systems

In this paper we demonstrate themethod of the enveloping first integral via an example of a completely integrable system of differential equations. This method allows a researcher to find and investigate singular invariant manifolds for a given family of invariant manifolds of steady motions represented by an initial system of equations. We describe specific properties of branching of the obtained families of singular invariant manifolds.     

Localization of invariant compact sets of families of discrete-time systems

The functional method of localization of invariant compact sets developed for continuous- and discrete-time dynamical systems is extended to families of discrete-time systems. Positively invariant compact sets are considered. As an example, the method is applied to the Hénon system with uncertain parameters.     

Number of Carleson sequences

This note is devoted to the refinement of a result of N. K. Nikol'skii and B. S. Pavlov (Ref. Zh. Mat. 1970, 6B678) describing the distribution of the discrete spectrum of a certain class of operators. Namely, one proves the following. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Mathematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 178–182, 1976.     

Cyclic invariant sets for one class of maps

We consider the dynamical system that is determined by a multidimensional map with scalar type nonlinearity and a nonnegative matrix of special form. For this map we establish the bifurcation character for the location of cyclic invariant sets in the phase space of the system, determine their location and periods depending on the properties of the matrix.     

Carleson Measures via BMO

We obtain new characterizations of Carleson measures via uniform boundedness of BMO norms of certain mass functions associated with the given measure in a natural way. This research was performed during M. Stessin’s visit to Korea University. He thanks the Mathematics Department of Korea University and the “Brain Pool” program for their hospitality and support. The first two authors were supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-314-C00012).     

On the Carleson duality

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their L ₂ Whitney averages belongs to L ₂ on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to L ₂ on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L _p generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.     

Carleson测度和多项式逼近

设G是复平面C上的单连通区域,在研究G上的Carleson测度和多项式在G上的Hardy空间Hq(G)中的稠密性问题时,建立起了两者之间的一种等价关系.用Carleson测度给出了多项式在Hq(G)中稠密的充要条件,也刻画了所有那些使得在其上的Carleson测度和Carleson不等式等价的单连通区域.     

Haar type and Carleson constants

For a collection of dyadic intervals, a Banach space X, andp(1, 2], we assume the upper ^p estimates where x_IX, and h_I denotes the L normalized Haar function supportedon I. We determine the minimal requirement on the size of suchthat these estimates imply that X is of Haar type p. The characterizationis given in terms of the Carleson constant of .