Means on solenoids

It is determined which solenoids admit 2-argument continuous means.

     

Homogeneous weak solenoids

A (generalized) weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps. If the covering maps are regular, then we call the inverse limit space a strong solenoid. By a theorem of M.C. McCord, strong solenoids are homogeneous. We show conversely that homogeneous weak solenoids are topologically equivalent to strong solenoids. We also give an example of a weak solenoid that has simply connected path-components, but which is not homogeneous.

     

3-manifolds that admit knotted solenoids as attractors

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.

     

Explicit representation of roots on <Emphasis Type="Italic">p</Emphasis>-adic solenoids and non-uniqueness of embeddability into rational one-parameter subgroups

This note generalizes known results concerning the existence of roots and embedding one-parameter subgroups on p-adic solenoids. An explicit representation of the roots leads to the construction of two distinct rational embedding one-parameter subgroups. The results contribute to enlighten the group structure of solenoids and to point out difficulties arising in the context of the embedding problem in probability theory. As a consequence, the uniqueness of embedding of infinitely divisible probability measures on p-adic solenoids is solved under a certain natural condition.     

A characterization of bilateral operator weighted shifts being Cowen-Douglas operators

It is characterized when a bilateral operator weighted shift is a Cowen-Douglas operator.

     

Hypercyclicity and supercyclicity for invertible bilateral weighted shifts

We give a characterization of the invertible bilateral weighted shifts that are hypercyclic or supercyclic. Although there is a general characterization due to H. Salas, in the invertible case the conditions simplify greatly.

     

Leibenzon's backward shift and composition operators

We apply Leibenzon's backward shift to show that the composition operator on the unit ball of always maps the weighted Hardy space into the Hardy class .

     

There are no piecewise linear maps of type

The aim of this paper is to show that there are no piecewise linear maps of type . For this purpose we use the fact that any piecewise monotone map of type has an infinite -limit set which is a subset of a doubling period solenoid. Then we prove that piecewise linear maps cannot have any doubling period solenoids.

     

On a conjecture of ridge

The conjecture of Ridge on the numerical range of a shift of periodic weights is resolved in the affirmative, i.e., if the weights are nonzero, the numerical range of the corresponding shift is an open disc centered at the origin. The radius of the disc can be expressed as the Perron root of a nonnegative irreducible symmetric matrix. Some related results are obtained.

     

The realization of Smale solenoid type attractors in 3-manifolds

We study how to realize Smale solenoid type attractors in 3-manifolds. It is already known that we can restrict the 3-manifolds to lens spaces. We get all Smale solenoids realized in a given lens space through an inductive construction. We turn this around to address the question of how to decide whether a closed braid is a trivial knot in S³. For a diffeomorphism f of a 3-manifold M that realizes a Smale solenoid, it is natural to ask whether f⁻¹ also realizes a Smale solenoid. We relate this question to exchangeable braids, and for some special positive case, we describe the relation between the two Smale solenoids of f and f⁻¹.     

No embeddings of solenoids into surfaces

A quick proof of Bing's theorem indicated by the title is given. Indeed the inverse limit of a sequence of degree maps between closed oriented -manifolds can never be embedded into any closed orientable -manifold. The proof also concludes Gumerov's result on the covering of solenoids.

     

Noninterpolatory Hermite subdivision schemes

Bivariate interpolatory Hermite subdivision schemes have recently been applied to build free-form subdivision surfaces. It is well known to geometric modelling practitioners that interpolatory schemes typically lead to ``unfair" surfaces--surfaces with unwanted wiggles or undulations--and noninterpolatory (a.k.a. approximating in the CAGD community) schemes are much preferred in geometric modelling applications. In this article, we introduce, analyze and construct noninterpolatory Hermite subdivision schemes, a class of vector subdivision schemes which can be applied to iteratively refine Hermite data in a not necessarily interpolatory fashion. We also study symmetry properties of such subdivision schemes which are crucial for application in free-form subdivision surfaces.

A key step in our mathematical analysis of Hermite type subdivision schemes is that we make use of the strong convergence theory of refinement equations to convert a prescribed geometric condition on the subdivision scheme--namely, the subdivision scheme is of Hermite type--to an algebraic condition on the subdivision mask. The latter algebraic condition can then be used in a computational framework to construct specific schemes.

     

The backward shift on the space of Cauchy transforms

This note examines the subspaces of the space of Cauchy transforms of measures on the unit circle that are invariant under the backward shift operator . We examine this question when the space of Cauchy transforms is endowed with both the norm and weak topologies.

     

Convergence of the shifted algorithm for unitary Hessenberg matrices

This paper shows that for unitary Hessenberg matrices the algorithm, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.

     

Convergence of the unitary algorithm with a unimodular Wilkinson shift

In applying the algorithm to compute the eigenvalues of a unitary Hessenberg matrix, a projected Wilkinson shift of unit modulus is proposed and proved to give global convergence with (at least) a quadratic asymptotic rate for the iteration. Experimental testing demonstrates that the unimodular shift produces more efficient numerical convergence.

     

Orthonormal wavelets and shift invariant generalized multiresolution analyses

All wavelets can be associated to a multiresolution-like structure, i.e. an increasing sequence of subspaces of . We consider the interaction of a wavelet and the shift operator in terms of which of the subspaces in this multiresolution-like structure are invariant under the shift operator. This action defines the notion of the shift invariance property of order . In this paper we show that wavelets of all levels of shift invariance exist, first for the classic case of dilation by 2, and then for arbitrary integral dilation factors.

     

Twist points of the von Koch snowflake

It is known that the set of twist points in the boundary of the von Koch snowflake domain has full harmonic measure. We provide a new, simple proof, based on the doubling property of the harmonic measure, and on the existence of an equivalent measure, invariant and ergodic with respect to the shift.

     

Hypercyclic operators that commute with the Bergman backward shift

The backward shift on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: ``Which operators that commute with inherit its hypercyclicity?' We show that the problem reduces to the study of operators of the form where is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question of hypercyclicity for such an operator depends on how freely is allowed to approach the unit circle as .

     

Nikolskii-type inequalities for shift invariant function spaces

Let be a vectorspace of complex-valued functions defined on of dimension over . We say that is shift invariant (on ) if implies that for every , where on . In this note we prove the following.

Theorem. Let be a shift invariant vectorspace of complex-valued functions defined on of dimension over . Let . Then

for every and

     

Operators with eigenvalues and extreme cases of stability

In the following, we consider some cases where the point spectrum of an operator is either very stable or very unstable with respect to small perturbations of the operator. The main result is about the shift operator on whose point spectrum is what we will call strongly stable. We also give some general perturbation results, including a result about the size of the set of operators that have an eigenvalue.