Continuous itinerary functions and dendrite maps

We examine the “kneading sequence” theory of maps on dendrites, concentrating on those maps having one “turning point” and the “unique itinerary property” (i.e., distinct points have distinct itineraries). This theory has major overlaps with the theories of polynomial Julia Sets and Hubbard Trees, but also has significant differences from those two theories (for which the unique itinerary property does not always hold). We show that the unique itinerary property is a powerful property, and allows a simple classification of such dendrite maps with respect to their kneading sequences (up to conjugacy if there is no nontrivial invariant subdendrite).One of the major tools introduced here is the continuous itinerary function. If one takes the set of all sequences of the symbols used to define the itineraries with respect to a partition of a space, there is a natural topology which forces the itinerary function (from the original space into the space of sequences of symbols) to be continuous, although this often leads to a non-Hausdorff itinerary topology. Despite this apparent drawback, we show that this itinerary topology is a useful tool for analyzing the dynamics of continuous maps on metric spaces.Characterizations in terms of kneading sequences are given for topological properties of these maps, including various transitivity properties, (in)decomposability of inverse limits, and the existence of certain “piecewise linearizations” of such maps which are a natural generalization of “tent” maps on the interval. The itinerary topology provides a natural topology for the parameter space of all kneading sequences, a natural subspace of which will be shown to have a one-point compactification that is a dendrite, with an interesting connection to the Mandelbrot Set.     

Quantum statistical mechanics over function fields

In this paper we construct a noncommutative space of “pointed Drinfeld modules” that generalizes to the case of function fields the noncommutative spaces of commensurability classes of Q-lattices. It extends the usual moduli spaces of Drinfeld modules to possibly degenerate level structures. In the second part of the paper we develop some notions of quantum statistical mechanics in positive characteristic and we show that, in the case of Drinfeld modules of rank one, there is a natural time evolution on the associated noncommutative space, which is closely related to the positive characteristic L-functions introduced by Goss. The points of the usual moduli space of Drinfeld modules define KMS functionals for this time evolution. We also show that the scaling action on the dual system is induced by a Frobenius action, up to a Wick rotation to imaginary time.     

Tangent Processes on Wiener Space

This paper deals with the study of the Malliavin calculus of Euclidean motions on Wiener space (i.e. transformations induced by general measure-preserving transformations, called “rotations”, and H-valued shifts) and the associated flows on abstract Wiener spaces.     

Vector-valued invariant means revisited

We show that a Banach space X is complemented in its ultraproducts if and only if for every amenable semigroup S the space of bounded X-valued functions defined on S admits (a) an invariant average; or (b) what we shall call “an admissible assignment”. Condition (b) still provides an equivalence for quasi-Banach spaces, while condition (a) necessarily implies that the space is locally convex.     

Functions of bounded variation on “good” metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincaré inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L¹ topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Carathéodory spaces.     

Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations

We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26].     

Stable manifolds for nonuniform polynomial dichotomies

We establish the existence of smooth stable manifolds in Banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy. This new dichotomy is more restrictive in the “nonuniform part” but allow the “uniform part” to obey a polynomial law instead of an exponential (more restrictive) law. We consider two families of perturbations. For one of the families we obtain local Lipschitz stable manifolds and for the other family, assuming more restrictive conditions on the perturbations and its derivatives, we obtain C¹ global stable manifolds. Finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family.     

Maximum Lebesgue extension of monotone convex functions

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a “nice” dual representation of the function.     

Holomorphe Störungstheorie in lokalkonvexen Räumen

In this paper we study bounded holomorphic perturbations of a semi-Fredholm operator between sequentially complete locally convex spaces; however, some results are new in the case of Banach spaces, too. We define a concept of holomorphy for bounded operator functions and show that a meromorphy theorem is true for such perturbations of the identity. Then we deal with the problem when a weakly holomorphic bounded operator function is holomorphic in the defined sense. In the case of one complex variable we then prove an existence and extension theorem for solutions of equations T(z)x=y(z) which answers a question of B. Gramsch [7]. Finally we apply our results to partial differential operators.     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-λ inequality with two parameters and the other uses Calderón-Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all Lp spaces for 1<p<∞. Pointwise estimates are then replaced by appropriate localized Lp-Lq estimates. We obtain weighted Lp estimates for a range of p that is different from (1,∞) and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.     

Hardy spaces that support no compact composition operators

We consider, for G a simply connected domain and 0<p<∞, the Hardy space formed by fixing a Riemann map τ of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of |F|^p over the curves τ({|z|=r}) be bounded for 0<r<1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is reflected in our Main Theorem: supports compact composition operators if and only if∂Ghas finite one-dimensional Hausdorff measure. Our work is inspired by an earlier result of Matache (Proc. Amer. Math. Soc. 127 (1999) 1483), who showed that the spaces of half-planes support no compact composition operators. Our methods provide a lower bound for the essential spectral radius which shows that the same result holds with “compact” replaced by “Riesz.” We prove similar results for Bergman spaces, with the Hardy-space condition “∂G has finite Hausdorff 1-measure” replaced by “G has finite area.” Finally, we characterize those domains G for which every composition operator on either the Hardy or the Bergman spaces is bounded.     

Remarks on some basic concepts in the KKM theory

In the KKM theory, some authors adopt the concepts of the compact closure (ccl), compact interior (cint), transfer compactly closed-valued multimap, transfer compactly l.s.c. multimap, and transfer compactly local intersection property, respectively, instead of the closure, interior, closed-valued multimap, l.s.c. multimap, and possession of a finite open cover property. In this paper, we show that such adoption is inappropriate and artificial. In fact, any theorem with a term with “transfer” attached is equivalent to the corresponding one without “transfer”. Moreover, we can invalidate terms with “compactly” attached by giving a finer topology on the underlying space. In such ways, we obtain simpler formulations of KKM type theorems, Fan-Browder type fixed point theorems, and other results in the KKM theory on abstract convex spaces.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

The Oshima-Sekiguchi and Liouville theorems on Heintze groups

Let L be an elliptic operator on a Riemannian manifold M. A function F annihilated by L is said to be L-harmonic. F is said to have moderate growth if and only if F grows at most exponentially in the Riemannian distance. If M is a rank-one symmetric space and L is the Laplace-Beltrami operator for M, the Oshima-Sekiguchi theorem [T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980) 1-81] states that a L-harmonic function F has moderate growth if and only if F is the Poisson integral of a distribution on the Furstenberg boundary. In this work we prove that this result generalizes to a very large class of homogeneous Riemannian manifolds of negative curvature. We also (i) prove a Liouville type theorem that characterizes the “polynomial-like” harmonic functions which vanish on the boundary in terms of their growth properties, (ii) describe all “polynomial-like” harmonic functions, and (iii) give asymptotic expansions for the Poisson kernel. One consequence of this work is that every Schwartz distribution on the boundary is the boundary value for a L-harmonic function F which is uniquely determined modulo “polynomial-like” harmonic functions.     

The localized longitudinal index theorem for Lie groupoids and the van Est map

We define the “localized index” of longitudinal elliptic operators on Lie groupoids associated with Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition of the “global index”, that can be computed by localization. This correspondence between the “global” and “localized” index is given by the van Est map for Lie groupoids.     

Factorizing τ-spectra of mappings

In this paper by a spectrum of mappings we mean a morphism of spectra of spaces. However, using the notion of a mapping of mappings, we give the definition of a spectrum of mappings similar to that of a spectrum of spaces. In this case, the formulations of the given results are also similar to the formulations of the corresponding results concerning the spectra of spaces.For the spectra of mappings we define the notion of a τ-spectrum of mappings factorizing in a special sense and prove a version of the Spectral Theorem for such spectra. Furthermore, to a given indexed collection F of mapping we associate a τ-spectrum factorizing in the above special sense whose mappings are Containing Mappings for F constructed in Iliadis (2005) [4]. These associated τ-spectra and the corresponding version of the Spectral Theorem imply that for a given indexed collection F of mappings any so-called “natural” τ-spectrum for F factorizing in the special sense contains a cofinal and τ-closed subspectrum whose mappings are Containing Mapping for F. Thus, Containing Mappigs for F appear here without any concrete construction. The associated τ-spectra are used also in order to define and characterize the so-called second-type saturated classes of mappings (which are “saturated” by universal elements).