On hereditarily indecomposable continua, Henderson compacta and a question of Yohe

We answer a question of Yohe by showing that there exists a family of continuum many topologically different hereditarily indecomposable Cantor manifolds without any non-trivial weakly infinite-dimensional subcontinua. This family may consist either of compacta containing one-dimensional subsets or of compacta containing no weakly infinite-dimensional subsets of positive dimension.

     

Hyperspaces and open monotone maps of hereditarily indecomposable continua

We prove the following theorems:

Theorem 1. Let be an -dimensional hereditarily indecomposable continuum. Then there exist -dimensional hereditarily indecomposable continua and monotone maps such that is an embedding and the space of all subcontinua of is embeddable in by .

Theorem 2. For every open monotone map with non-trivial sufficiently small fibers on a finite dimensional hereditarily indecomposable continuum with there exists a -dimensional subcontinuum such that and the restriction of to is also monotone and open.

The connection between these theorems and other results in Hyperspace theory is studied.

     

Strictly singular non-compact operators on hereditarily indecomposable Banach spaces

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic Banach space. The construction of this operator relies on the existence of transfinite -spreading models in the dual of the space.

     

Interpolating hereditarily indecomposable Banach spaces

The following dichotomy is proved.

Every Banach space either contains a subspace isomorphic to , or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space.

In the particular case of , it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator admits a factorization through a H.I. space. The same result holds for every strictly singular operator .

Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.     

Exactly -to-1 maps and hereditarily indecomposable tree-like continua

In 1947, W.H. Gottschalk proved that no dendrite is the continuous, exactly -to-1 image of any continuum if . Since that time, no other class of continua has been shown to have this same property. It is shown that no hereditarily indecomposable tree-like continuum is the continuous, exactly -to-1 image of any continuum if .

     

A hereditarily indecomposable tree-like continuum without the fixed point property

A hereditarily indecomposable tree-like continuum without the fixed point property is constructed. The example answers a question of Knaster and Bellamy.

     

A universal property of reflexive hereditarily indecomposable Banach spaces

It is shown that every separable Banach space universal for the class of reflexive Hereditarily Indecomposable space contains isomorphically and hence it is universal for all separable spaces. This result shows the large variety of reflexive H.I. spaces.

     

The Space of subcontinua of a 2-dimensional continuum is infinite dimensional

Let be a metric continuum and let denote the space of subcontinua of with the Hausdorff metric. We settle a longstanding problem showing that if then . The special structure and properties of hereditarily indecomposable continua are applied in the proof.

     

A generalization of the cellular indecomposable property via fiber dimension

The cellular indecomposable property, introduced by Olin and Thomson in 1984 [11], is well known for the Dirichlet space, but it fails trivially for the vector-valued case. The purpose of this paper is to use the fiber dimension to reformulate the property such that it naturally extends the scalar-valued case, yet fix the vector-valued case in a meaningful way. Using the new formulation, we are able to generalize several previous results to the vector-valued setting. In particular, we extend a theorem of Bourdon relating the cellular indecomposable property and the codimension-one property to codimension-N. Several of our results appear to be new even for the Hardy space over the unit disc.     

Homogeneous continua in 2-manifolds

In 1960 R.H. Bing [2] proved that every homogeneous plane continuum that contains an arc is a simple closed curve. At that time Bing [2, p. 228] asked if every 1-dimensional homogeneous continuum that contains an arc and lies on a 2-manifold is a simple closed curve. We prove that no 2-manifold contains uncountably many disjoint triods. We use this theorem and decomposition theorems of F.B. Jones [10] and H.C. Wiser [19] to answer Bing's question in the affirmative. We also prove that every homogeneous indecomposable continuum in a 2-manifold can be embedded in the plane. It follows from this result and another theorem of Wiser [20] that every homogeneous continuum that is properly contained in an orientable 2-manifold is planar.     

Uniqueness of complex structure and real hereditarily indecomposable Banach spaces

There exists a real hereditarily indecomposable Banach space X=X(C) (respectively X=X(H)) such that the algebra L(X)/S(X) is isomorphic to C (respectively to the quaternionic division algebra H).Up to isomorphism, X(C) has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable. So there exist two Banach spaces which are isometric as real spaces but totally incomparable as complex spaces. This extends results of J. Bourgain and S. Szarek [J. Bourgain, Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (2) (1986) 221-226; S. Szarek, On the existence and uniqueness of complex structure and spaces with “few” operators, Trans. Amer. Math. Soc. 293 (1) (1986) 339-353; S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437-444], and proves that a theorem of G. Godefroy and N.J. Kalton [G. Godefroy, N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (1) (2003) 121-141] about isometric embeddings of separable real Banach spaces does not extend to the complex case.The quaternionic example X(H), on the other hand, has unique complex structure up to isomorphism; other examples with a unique complex structure are produced, including a space with an unconditional basis and non-isomorphic to l₂. This answers a question of S. Szarek in [S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437-444].     

The Hausdorff dimension of visible sets of planar continua

For a compact set and a point , we define the visible part of from to be the set

(Here denotes the closed line segment joining to .)

In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly less than the Hausdorff dimension of . In fact, for almost every ,

We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than for some .

     

不可分解模的参量化

本文的主要结果是给出了有限群的不可分解模的一种参量化, 它相对于Puig 和Thévenaz 给出的参量化更加自然. 同时我们说明了用这两种不同的方式给出的参量化是一致的.     

On the Indecomposable Modules of Uq（sl（2）） Constructed via Ore-extension

Let A be a subalgebra of Uq (sl(2)) generated by K, K-1 and F and Aδ be a subalgebra of Uq (sl(2)) generated by K, K-1 (and also Fd if q is a primitive d-th root of unity with d an odd number). Given an Aδ -module M, a Uq (sl(2))-module AAδ M is constructed via the iterated Ore extension of Uq (sl(2)) in a unified framework for any q. Then all the submodules of AAδ M are determined for a fixed finite-dimensional indecomposable Aδ -module M . It turns out that for some indecomposable Aδ -module M , the Uq (sl(2))-module AAδ M is indecomposable, which is not in the BGG-categories Oq associated with quantum groups in general.     

On the existence of a finite invariant measure

A necessary and sufficient condition is given for a Borel automorphism on a standard Borel space to admit an invariant probability measure.     

A theorem on planar continua and an application to automorphisms of the field of complex numbers

Let K be a continuum in the plane which does not lie on a line. Then the set of differences, K - K, contains an open set. Let ψ be an automorphism of the field of complex numbers which is bounded on an F_σ set of positive inductive dimension. Then ψ is continuous.     

On the dimension of a homeomorphism group

We prove that the homeomorphism group of each one of a collection of continua constructed in a paper by the first author (Trans. Amer. Math. Soc. 121 (1966), 516-548) is one dimensional. This answers a question posed in that paper.

     

Rational irreducible plane continua without the fixed-point property

We define rational irreducible continua in the plane that admit fixed-point-free maps with the condition that all of their tranches have the fixed-point property. This answers in the affirmative a question of Hagopian. The construction is based on a special class of spirals that limit on a double Warsaw circle. The closure of each of these spirals has the fixed-point property.