Geometry and ergodic theory of non-hyperbolic exponential maps

We deal with all the maps from the exponential family such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters are included. We introduce as our main technical devices the projection of the map to the infinite cylinder and an appropriate conformal measure . We prove that , essentially the set of points in returning infinitely often to a compact region of disjoint from the orbit of , has the Hausdorff dimension , that the -dimensional Hausdorff measure of is positive and finite, and that the -dimensional packing measure is locally infinite at each point of . We also prove the existence and uniqueness of a Borel probability -invariant ergodic measure equivalent to the conformal measure . As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the -limit set (under ) of Lebesgue almost every point in , coincides with the orbit of zero under the map . Finally we show that the the function , , is continuous.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

Complex symmetric operators and applications II

A bounded linear operator on a complex Hilbert space is called complex symmetric if , where is a conjugation (an isometric, antilinear involution of ). We prove that , where is an auxiliary conjugation commuting with . We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compressed shift). The decomposition also extends to the class of unbounded -selfadjoint operators, originally introduced by Glazman. In this context, it provides a method for estimating the norms of the resolvents of certain unbounded operators.

     

Torsion freeness of symmetric powers of ideals

Let be an ideal in a Noetherian commutative ring with unit, let be an integer, and let be the canonical surjective -module homomorphism from the th symmetric power of to the th power of . When or when is a perfect Gorenstein ideal of grade , we provide a necessary and sufficient condition for to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of . When is a maximal ideal of we show that is an isomorphism if and only if is a regular local ring. In all three cases for our results yield that if is an isomorphism, then is also an isomorphism for each .

     

On homeomorphic Bernoulli measures on the Cantor space

Let be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights and . It is a long-standing open problem to characterize those and such that and are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending to ). The (possibly) weaker property of and being continuously reducible to each other is equivalent to a property of and called binomial equivalence. In this paper we define an algebraic property called ``refinability' and show that, if and are refinable and binomially equivalent, then and are topologically equivalent. Next we show that refinability is equivalent to a fairly simple algebraic property. Finally, we give a class of examples of binomially equivalent and refinable numbers; in particular, the positive numbers and such that and are refinable, so the corresponding measures are topologically equivalent.

     

Quadratic maps and Bockstein closed group extensions

Let be a central extension of the form where and are elementary abelian -groups. Associated to there is a quadratic map , given by the -power map, which uniquely determines the extension. This quadratic map also determines the extension class of the extension in and an ideal in which is generated by the components of . We say that is Bockstein closed if is an ideal closed under the Bockstein operator.

We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map given by yield Bockstein closed extensions.

On the other hand, it is well known that an extension is Bockstein closed if and only if it lifts to an extension for some -lattice . In this situation, one may write for a ``binding matrix' with entries in . We find a direct way to calculate the module structure of in terms of . Using this, we study extensions where the lattice is diagonalizable/triangulable and find interesting equivalent conditions to these properties.

     

Semi-edges, reflections and Coxeter groups

We combine the theory of Coxeter groups, the covering theory of graphs introduced by Malnic, Nedela and Skoviera and the theory of reflections of graphs in order to obtain the following characterization of a Coxeter group:

Let be a -covering of a monopole admitting semi-edges only. The graph is the Cayley graph of a Coxeter group if and only if is regular and any deck transformation in that interchanges two neighboring vertices of acts as a reflection on .

     

Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type

We establish the uniqueness of the positive solution for equations of the form in , . The special feature is to consider nonlinearities whose variation at infinity is not regular (e.g., , , , , , , or ) and functions in vanishing on . The main innovation consists of using Karamata's theory not only in the statement/proof of the main result but also to link the nonregular variation of at infinity with the blow-up rate of the solution near .

     

Algebraic hypergeometric transformations of modular origin

It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function  arises from a relation between modular curves, namely the covering of  by . In general, when  , the -fold cover of  by  gives rise to an algebraic hypergeometric transformation. The transformations are arithmetic-geometric mean iterations, but the transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since are of genus . Since their quotients under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.

     

The distance function from the boundary in a Minkowski space

Let the space be endowed with a Minkowski structure (that is, is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class ), and let be the (asymmetric) distance associated to . Given an open domain of class , let be the Minkowski distance of a point from the boundary of . We prove that a suitable extension of to (which plays the rôle of a signed Minkowski distance to ) is of class in a tubular neighborhood of , and that is of class outside the cut locus of (that is, the closure of the set of points of nondifferentiability of in ). In addition, we prove that the cut locus of has Lebesgue measure zero, and that can be decomposed, up to this set of vanishing measure, into geodesics starting from and going into along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point outside the cut locus the pair , where denotes the (unique) projection of on , and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

     

Analyticity on translates of a Jordan curve

Let be a domain in which is symmetric with respect to the real axis and whose boundary is a real analytic simple closed curve. Translate vertically to get where is such that . We prove that if is a continuous function on such that for each , the function has a continuous extension to which is holomorphic on , then is holomorphic on .

     

Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

In the first section of this paper we revisit the definition and some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra over an arbitrary field . Our main observation is that , the minimal polynomial of , may depend not only on , but also on the underlying algebra. More precisely, if is a subalgebra of , and if is the minimal polynomial of in , then may differ from , in which case we have .

In the second section we restrict attention to the case where is either the real or the complex numbers, and define , the radius of an element in , to be the largest root in absolute value of the minimal polynomial of . We show that possesses some of the familiar properties of the classical spectral radius. In particular, we prove that is a continuous function on .

In the third and last section, we deal with stability of subnorms acting on subsets of finite-dimensional power-associative algebras. Following a brief survey, we enhance our understanding of the subject with the help of our findings of the previous section. Our main new result states that if , a subset of an algebra , satisfies certain assumptions, and is a continuous subnorm on , then is stable on if and only if majorizes the radius defined above.

     

Equivalence of domains arising from duality of orbits on flag manifolds III

In Gindikin and Matsuki 2003, we defined a - invariant subset of for each -orbit on every flag manifold and conjectured that the connected component of the identity would be equal to the Akhiezer-Gindikin domain if is of nonholomorphic type. This conjecture was proved for closed in Wolf and Zierau 2000 and 2003, Fels and Huckleberry 2005, and Matsuki 2006 and for open in Matsuki 2006. It was proved for the other orbits in Matsuki 2006, when is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed -orbit when is of Hermitian type. Thus the conjecture is completely solved affirmatively.

     

Extensions of -local finite groups

A -local finite group consists of a finite -group , together with a pair of categories which encode ``conjugacy' relations among subgroups of , and which are modelled on the fusion in a Sylow -subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as -completed classifying spaces of finite groups. In this paper, we study and classify extensions of -local finite groups, and also compute the fundamental group of the classifying space of a -local finite group.

     

Asymptotic distribution of the largest off-diagonal entry of correlation matrices

Suppose that we have observations from a -dimensional population. We are interested in testing that the variates of the population are independent under the situation where goes to infinity as . A test statistic is chosen to be , where is the sample correlation coefficient between the -th coordinate and the -th coordinate of the population. Under an independent hypothesis, we prove that the asymptotic distribution of is an extreme distribution of type , by using the Chen-Stein Poisson approximation method and the moderate deviations for sample correlation coefficients. As a statistically more relevant result, a limit distribution for , where is Spearman's rank correlation coefficient between the -th coordinate and the -th coordinate of the population, is derived.

     

Differentiability of spectral functions for symmetric -stable processes

Let be a signed Radon measure in the Kato class and define a Schrödinger type operator on . We show that its spectral bound is differentiable if and is Green-tight.

     

Singular solutions of parabolic -Laplacian with absorption

We consider, for and , the -Laplacian evolution equation with absorption

We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in , and satisfy for all . We prove the following:

(i)

When , there does not exist any such singular solution.

(ii)

When , there exists, for every , a unique singular solution that satisfies as .

Also, as , where is a singular solution that satisfies as .

Furthermore, any singular solution is either or for some finite positive .

     

On embeddings of proper and equicontinuous actions in zero-dimensional compactifications

We provide a tool for studying properly discontinuous actions of non-compact groups on locally compact, connected and paracompact spaces, by embedding such an action in a suitable zero-dimensional compactification of the underlying space with pleasant properties. Precisely, given such an action we construct a zero-dimensional compactification of with the properties: (a) there exists an extension of the action on , (b) if is the set of the limit points of the orbits of the initial action in , then the restricted action remains properly discontinuous, is indivisible and equicontinuous with respect to the uniformity induced on by that of , and (c) is the maximal among the zero-dimensional compactifications of with these properties. Proper actions are usually embedded in the endpoint compactification of , in order to obtain topological invariants concerning the cardinality of the space of the ends of , provided that has an additional ``nice" property of rather local character (``property Z", i.e., every compact subset of is contained in a compact and connected one). If the considered space has this property, our new compactification coincides with the endpoint one. On the other hand, we give an example of a space not having the ``property Z" for which our compactification is different from the endpoint compactification. As an application, we show that the invariant concerning the cardinality of the ends of holds also for a class of actions strictly containing the properly discontinuous ones and for spaces not necessarily having ``property Z".