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Given a coarse space $(X,\mathcal{E})$, one can define a $\mathrm{C}^*$-algebra $\mathrm{C}^*_u(X)$ called the uniform Roe algebra of $(X,\mathcal{E})$. It has been proved by J. \v{S}pakula and R. Willett that if the uniform Roe algebras of two uniformly locally finite metric spaces with property A are isomorphic, then the metric spaces are coarsely equivalent to each other. In this paper, we look at the problem of generalizing this result for general coarse spaces and on weakening the hypothesis of the spaces having property A.

We prove a generalization of the fundamental inequality of Guivarc'h relating entropy, drift and critical exponent to Gibbs measures on geometrically finite quotients of CAT(-1) metric spaces. For random walks with finite superexponential moment, we show that the equality is achieved if and only if the Gibbs density is equivalent to the hitting measure. As a corollary, if the action is not convex cocompact, any hitting measure is singular to any Gibbs density., Comment: 29 pages

Let G be the homeomorphism group of a dendrite. We study the normal subgroups of G. For instance, there are uncountably many non-isomorphic such groups G that are simple groups. Moreover, these groups can be chosen so that any isometric G-action on any metric space has a bounded orbit. In particular they have the fixed point property (FH)., Comment: Modifications according to referee's comments

The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of non-linear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with some Besov-type topology. The proofs are based on atomic decomposition techniques and metric embeddings. Additionally, we provide embedding results showing how non-linear Besov spaces embed into non-linear $p$-variation spaces and vice versa. We emphasize that we neither assume the UMD property of the involved spaces nor their separability., 21 pages

The equivalences to and the connections between the modulus-of-continuity condition, compact containment and tightness on D E [ a , b ] D_{E}[a,b] with a > b a>b are studied. The results within are tools for establishing tightness for probability measures on D E [ a , b ] D_E[a,b] that generalize and simplify prevailing results in the cases that E E is a metric space, nuclear space dual or, more generally, a completely regular topological space. Applications include establishing weak convergence to martingale problems, the long-time typical behavior of nonlinear filters and particle approximation of cadlag probability-measure-valued processes. This particle approximation is studied herein, where the distribution of the particles is the underlying measure-valued process at an arbitrarily fine discrete mesh of points.

We introduce the quantum Gromov-Hausdorff propinquity, a new distance between quantum compact metric spaces, which extends the Gromov-Hausdorff distance to noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff distance and Rieffel's proximity by making *-isomorphism a necessary condition for distance zero, while being well adapted to Leibniz seminorms. This work offers a natural solution to the long-standing problem of finding a framework for the development of a theory of Leibniz Lip-norms over C*-algebras., Comment: 49 Pages. This is the first half of 1302.4058v2, which has been accepted in Trans. Amer. Math. Soc. The second half is now a different paper entitled "Convergence of Fuzzy Tori and Quantum Tori for the quantum Gromov-Hausdorff Propinquity: an explicit approach"

In this paper, we prove that the quasihyperbolic metrics are quasiinvariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of above theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping.

The present paper is devoted to gradient flow semigroups of convex functionals on Hadamard spaces. We show that the Mosco convergence of a sequence of convex lsc functions implies the convergence of the corresponding resolvents and the convergence of the gradient flow semigroups. This extends the classical results of Attouch, Brezis and Pazy into spaces with no linear structure. The same method can be further used to show the convergence of semigroups on a sequence of spaces, which solves a problem of Kuwae and Shioya [Trans. Amer. Math. Soc. 360, no. 1, 2008].

A metric measure space is a complete separable metric space equipped with probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov-Prohorov metric of Greven, Pfaffelhuber and Winter. We consider the natural binary operation $\boxplus$ on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup with a translation invariant metric and that each element has a unique factorization into prime elements. We investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric. For example, we show that for any given positive real numbers $a,b,c$ the trivial space is the only space $\mathcal{X}$ that satisfies $a \mathcal{X} \boxplus b \mathcal{X} = c \mathcal{X}$. We establish that there is no analogue of the law of large numbers: if $\mathbf{X}_1, \mathbf{X}_2$..., is an identically distributed independent sequence of random spaces, then no subsequence of $\frac{1}{n} \boxplus_{k=1}^n \mathbf{X}_k$ converges in distribution unless each $\mathbf{X}_k$ is almost surely equal to the trivial space. We characterize the infinitely divisible probability measures and the L\'evy processes on this semigroup, characterize the stable probability measures and establish a counterpart of the LePage representation for the latter class., Comment: 48 pages, 0 figures. The previous version considered only compact metric measure spaces, but new arguments allow all the results of the previous version to be extended to the setting of complete separable metric measure spaces

A new notion of a viscosity solution for Eikonal equations in a general metric space is introduced. A comparison principle is established. The existence of a unique solution is shown by constructing a value function of the corresponding optimal control theory. The theory applies to infinite dimensional setting as well as topological networks, surfaces with singularities.

Given a locally compact Polish space X, a necessary and sufficient condition for a group G of homeomorphisms of X to be the full isometry group of (X,d) for some proper metric d on X is given. It is shown that every locally compact Polish group G acts freely on GxY as the full isometry group of GxY with respect to a certain proper metric on GxY, where Y is an arbitrary locally compact Polish space with (card(G),card(Y)) different from (1,2). Locally compact Polish groups which act effectively and almost transitively on complete metric spaces as full isometry groups are characterized. Locally compact Polish non-Abelian groups on which every left invariant metric is automatically right invariant are characterized and fully classified. It is demonstrated that for every locally compact Polish space X having more than two points the set of proper metrics d such that Iso(X,d) = {id} is dense in the space of all proper metrics on X., Comment: 24 pages

This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms. We call these points synchronization points. The resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces, toral automorphisms and compact abelian group epimorphisms. Synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered. Furthermore, these systems reveal strong connections between the new invariants and problems of active interest in number theory, relating to heights and greatest common divisors

Let $p$ be an even positive integer and $U_p(H)$ be the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. Let ${\cal O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow ${\cal O}$ with a natural Finsler metric in terms of the $p$-Schatten norm and the action of $U_p(H)$. Our main result establishes that for any pair of given initial conditions $$ x\in {\cal O}\hbox{and} X\in (T{\cal O})_x $$ there exists a curve $\delta(t)=e^{tz}\cdot x$ in ${\cal O}$, with $z$ a skew-hermitian element in the $p$-Schatten class such that $$ \delta(0)=x \hbox{and} \dot{\delta}(0)=X, $$ which remains minimal as long as $t\|z\|_p\le \pi/4$. Moreover, $\delta$ is unique with these properties. We also show that the metric space $({\cal O},d)$ ($d=$ rectifiable distance) is complete. In the process we establish minimality results in the groups $U_p(H)$, and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit $$ {\cal O}=\{uAu^*: u\in U_p(H)\} $$ of a self-adjoint operator $A\in B(H)$., Comment: 25 pages

We discuss and compare the notions of ideal boundaries, Floyd boundaries and Gromov boundaries of metric spaces. The three types of boundaries at infinity are compared in the general setting of unbounded length spaces as well as in the special cases of CAT(0) and Gromov hyperbolic spaces. Gromov boundaries, usually defined only for Gromov hyperbolic spaces, are extended to arbitrary metric spaces.

We show that if X is a proper metric measure space equipped with a doubling measure supporting a Poincare inequality, then subsets of X with zero p-capacity are precisely the p-polar sets; that is, a relatively compact subset of a domain in X is of zero p-capacity if and only if there exists a p-superharmonic function whose set of singularities contains the given set. In addition, we prove that if X is a p-hyperbolic metric space, then the p-superharmonic function can be required to be p-superharmonic on the entire space X. We also study the the following question: If a set is of zero p-capacity, does there exist a p-superharmonic function whose set of singularities is precisely the given set?

Let ( X , d ) (X,d) be a metric space. For a probability measure μ \mu on a subset E E of X X and a Vitali cover V V of E E , we introduce the notion of a b μ b_{\mu } -Vitali subcover V μ V_{\mu } , and compare the Hausdorff measures of E E with respect to these two collections. As an application, we consider graph directed self-similar measures μ \mu and ν \nu in R d \mathbb {R}^{d} satisfying the open set condition. Using the notion of pointwise local dimension of μ \mu with respect to ν \nu , we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen’s multifractal Hausdorff measures are mutually singular.

The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel′ on compactness of saddle surfaces in a Euclidean space.

While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure μ \mu is the countable dense subset { μ ( U ) : U \{ \mu (U) : U is clopen } \} of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure μ \mu is good if whenever U , V U, V are clopen sets with μ ( U ) > μ ( V ) \mu (U) > \mu (V) , there exists W W a clopen subset of V V such that μ ( W ) = μ ( U ) \mu (W) = \mu (U) . These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, G δ G_{\delta } conjugacy class.

Let (X, d) be a metric space, and f : X → X be a continuous map. In this paper we prove that if R(f) is compact, and w(x, f) ¬= O for all x ∈ X, then f is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism h and a non-expanding map g that is pointwise convergent to a fixed point v 0 such that f is uniformly conjugate to a subsystem (h x g)|S of the product map h x g. In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.

Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.

The determinant of the Ricci endomorphism of a Kähler metric is called its central curvature, a notion well-defined even in the Riemannian context. This work investigates two types of Kähler metrics in which this curvature potential gives rise to a potential for a gradient holomorphic vector field. These metric types generalize the Kähler-Einstein notion as well as that of Bando and Mabuchi (1986). Whenever possible the central curvature is treated in analogy with the scalar curvature, and the metrics are compared with the extremal Kähler metrics of Calabi. An analog of the Futaki invariant is employed, both invariants belonging to a family described in the language of holomorphic equivariant cohomology. It is shown that one of the metric types realizes the minimum of an L 2 L^2 functional defined on the space of Kähler metrics in a given Kähler class. For metrics of constant central curvature, results are obtained regarding existence, uniqueness and a partial classification in complex dimension two. Consequently, on a manifold of Fano type, such metrics and Kähler-Einstein metrics can only exist concurrently. An existence result for the case of non-constant central curvature is stated, and proved in a sequel to this work.

We prove the following embedding theorems in the coarse geometry: Theorem A. Every metric space X with bounded geometry whose asymptotic dimension does not exceed n admits a large scale uniform embedding into the product of n + 1 locally finite trees. Corollary. Every metric space X with bounded geometry whose asymptotic dimension does not exceed n admits a large scale uniform embedding into a non-positively curved manifold of dimension 2n + 2. The Corollary is used in the proof of the following. Theorem B. For every uniformly contractible manifold X whose asymptotic dimension is finite, the product X x R is integrally hyperspherical for some n. Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature. We also prove that if a uniformly contractible manifold X of bounded geometry is large scale uniformly embeddable into a Hilbert space, then X is stably integrally hyperspherical.

Let N be an isolating neighborhood for a map f. If we can decompose N into the disjoint union of compact sets N 1 and N 2 , then we can relate the dynamics on the maximal invariant set Inv N to the shift on two symbols by noting which component of N each iterate of a point x ∈ Inv N lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to N. In essence, we measure the difference between the Conley index of Inv N and the sum of the indices of Inv N 1 and Inv N 2 .

In 1948 Miroslav Katětov showed that if the cube X 3 X^{3} of a compact space X X satisfies the separation axiom T 5 _{5} then X X must be metrizable. He asked whether X 3 X^{3} can be replaced by X 2 X^{2} in this metrization result. In this note we prove the consistency of this implication.

Let{fn}\{f_{n} \}be a sequence of nonlinear operators. We discuss the asymptotic properties of their inhomogeneous iteratesfn∘fn−1∘⋯∘f1f_{n} \circ f_{n-1} \circ \cdots \circ f_{1}\,in metric spaces, then apply the results to the ordered Banach spaces through projective metrics. Theorems on path stability and nonlinear weak ergodicity are obtained in this paper.

This paper involves the generalization of minimal surface theory to spaces with singularities. Let X X be an NPC space, i.e. a metric space of non-positive curvature. We define a (parametric) minimal surface in X X as a conformal energy minimizing map. Using this definition, many properties of classical minimal surfaces can also be observed for minimal surfaces in this general setting. In particular, we will prove the boundary monotonicity property and the isoperimetric inequality for minimal surfaces in X X .

We show that if the universal covering of a compact Riemannian manifold with no conjugate points is a quasi-convex metric space then the following assertion holds: Either the universal covering of the manifold is a hyperbolic geodesic space or it contains a quasi-isometric immersion of Z × Z Z\times Z .

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolik for characterizing Cechcomplete spaces. We also show that the absolute Borel class of X is determined by the uniform structure of the space of continuous functions Cp (X); however the case of absolute G6 metric spaces is still open. More precisely, we prove that, for metrizable spaces X and Y, if 1, then Y is also an absolute Borel set of the same class. This result is new even if P is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the Cech-completeness of a metric space X is determined by the linear structure of Cp(X).

We give descriptions of the spaces D ( K ) D(K) (i.e. the space of differences of bounded semicontinuous functions on K K ) and especially of B 1 / 4 ( K ) B_{1/4}(K) (defined by Haydon, Odell and Rosenthal) as well as for the norms which are defined on them. For example, it is proved that a bounded function on a metric space K K belongs to B 1 / 4 ( K ) B_{1/4}(K) if and only if the ω t h \omega ^{ \mathrm {th}} -oscillation, o s c ω f \mathrm {osc}_{\omega }f , of f f is bounded and in this case ‖ f ‖ 1 / 4 = ‖ | f | + o s c ~ ω f ‖ ∞ \| f\|_{1/4}=\|\, |f|+ \widetilde {\mathrm {osc}}_{\omega } f\|_{\infty } . Also, we classify B 1 / 4 ( K ) B_{1/4}(K) into a decreasing family ( S ξ ( K ) ) 1 ≤ ξ > ω 1 (S_{\xi }(K))_{1\leq \xi >\omega _1} of Banach spaces whose intersection is equal to D ( K ) D(K) and S 1 ( K ) = B 1 / 4 ( K ) S_1 (K)=B_{1/4}(K) . These spaces are characterized by spreading models of order ξ \xi equivalent to the summing basis of c 0 c_0 , and for every function f f in S ξ ( K ) S_{\xi }(K) it is valid that o s ω ξ f \mathrm {os}_{\omega ^{\xi }}f is bounded. Finally, using the notion of null-coefficient of order ξ \xi sequence, we characterize the Baire-1 functions not belonging to S ξ ( K ) S_{\xi }(K) .

We determine the Hausdorff dimension of level sets and of sets of points of multiplicity for mappings in a residual subset of the space of all continuous mappings from Rn to Rm . Questions about the structure of level sets of typical (i.e. of all except a set negligible from the point of view of Baire category) real functions on the unit interval were already studied in [2], [1], and also [3]. In the last one the question about the Hausdorff dimension of level sets of such functions appears. The fact that a typical continuous function has all level sets zero-dimensional was commonly known, although it seems difficult to find the first proof of this. The author showed in [9] that in certain spaces of functions the level sets are of dimension one typically and developed in [8] a method to show that in other spaces functions are typically injective on the complement of a zero dimensional set-this yields smallness of all level sets. Here we are going to extend this method to continuous mappings between Euclidean spaces and to determine the size of level sets of typical mappings and the typical multiplicity in case the level sets are finite; see Theorems 1 and 2. We will use the following notation. For M a set and e > 0, U(M, e), resp. B(M, e), is the open, resp. closed, e-neighborhood of M and we write U(x, e) instead of U({x}, e) . Given a metric space X of functions, we say that typical f E X has property P if {f E X; non P(f)} is first category in X. We will always deal with spaces of continuous mappings equipped with the supremum metric. Finally, to prove that certain properties are typical, we will use the Banach-Mazur game. Let (X, p) be a metric space and M c X. In the first step Player A selects an open ball U(x1, e1) . In the second step Player B selects an open ball U(x2, 62) C U(xl, 61) and then A continues with U(x3, 83) C U(x2, 82), and so on. By definition Player B wins if Il B(xj, ej) c M, else Player A wins. We have the following There is a winning strategy for Player B if and only if X\M is a first category set in X. Received by the editors January 5, 1994; originally communicated to the Proceedings of theAMS by Andrew Bruckner. 1991 Mathematics Subject Classification. Primary 46E1 5, 26B99, 28A78.

Let X X and Y Y be metric spaces and let ϕ : C p ( X ) → C p ( Y ) \phi :{C_p}(X) \to {C_p}(Y) (resp. ϕ : C p ∗ ( X ) → C p ∗ ( Y ) \phi :C_p^\ast (X) \to C_p^\ast (Y) ) be a continuous linear surjection. We prove that Y Y is completely metrizable whenever X X is. As a corollary we obtain that complete metrizability is preserved by l p {l_p} (resp. l p ∗ l_p^\ast -equivalence) in the class of all metric spaces. This solves Problem 35 in [2] (raised by Arhangel’skiĭ).

We present a new concept of chaos, ω \omega -chaos, and prove some properties of ω \omega -chaos. Then we prove that ω \omega -chaos is equivalent to positive entropy on the interval. We also prove that ω \omega -chaos is equivalent to the definition of chaos given by Devaney on the interval.

Let T T be a metric space and X X a Banach space. Let F : T → X F:T \to X be a set-valued map assuming arbitrary values and satisfying the upper semicontinuity condition: { t ∈ T : F ( t ) ∩ C ≠ ∅ } \{ t \in T:F(t) \cap C \ne \emptyset \} is closed for each weakly closed set C C in X X . Then there is a sequence of norm-continuous functions converging pointwise (in the norm) to a selection for F F . We prove a statement of similar precision and generality when X X is a metric space.

An R R -tree is a metric space in which any two points are joined by a unique arc. Every arc is isometric to a closed interval of R R . When a group G G acts on a tree ( Z Z -tree) X X without inversion, then X / G X/G is a graph. One gets a presentation of G G from the quotient graph X / G X/G with vertex and edge stabilizers attached. For a general R R -tree X X , there is no such combinatorial structure on X / G X/G . But we can still ask what the maximum number of orbits of branch points of free actions on R R -trees is. We prove the finiteness of the maximum number for a family of groups, which contains free products of free abelian groups and surface groups, and this family is closed under taking free products with amalgamation.

On a compact surface with smooth boundary, the determinant of the Laplacian associated to a smooth metric on the surface (with Dirichlet boundary conditions if the boundary is nonempty) is a well-defined isospectral invariant. As a function on the moduli space of such surfaces, it is a smooth function whose boundary behavior in certain cases is well understood; see [OPS and K]. In this paper, we restrict ourselves to a certain class of singular metrics on closed surfaces called conical metrics. We show that the determinant of the associated Laplacian is still well defined and that it is a real analytic function on a suitably restricted subset of the space of conical metrics on the sphere.

We endow families of nonempty closed subsets of a metric space with uniformities defined by semimetrics. Such structure is completely determined by a class (which is a family of closed sets) and a type (which is a semimetric). Two types are sufficient to define (and classify) almost all convergences known till now. These two types offer the possibility of defining other set convergences.

R {\mathbf {R}} -trees arise naturally in the study of groups of isometries of hyperbolic space. An R {\mathbf {R}} -tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an R {\mathbf {R}} -tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize R {\mathbf {R}} -trees among metric spaces. A universal R {\mathbf {R}} -tree would be of interest in attempting to classify the actions of groups of isometries on R {\mathbf {R}} -trees. It is easy to see that there is no universal R {\mathbf {R}} -tree. However, we show that there is a universal separable R {\mathbf {R}} -tree T ℵ 0 {T_{{\aleph _0}}} . Moreover, for each cardinal α , 3 ≤ α ≤ ℵ 0 \alpha ,3 \leq \alpha \leq {\aleph _0} , there is a space T α ⊂ T ℵ 0 {T_\alpha } \subset {T_{{\aleph _0}}} , universal for separable R {\mathbf {R}} -trees, whose order of ramification is at most α \alpha . We construct a universal smooth dendroid D D such that each separable R {\mathbf {R}} -tree embeds in D D ; thus, has a smooth dendroid compactification. For nonseparable R {\mathbf {R}} -trees, we show that there is an R {\mathbf {R}} -tree X α {X_\alpha } , such that each R {\mathbf {R}} -tree of order of ramification at most α \alpha embeds isometrically into X α {X_\alpha } . We also show that each R {\mathbf {R}} -tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of R {\mathbf {R}} -trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.

We show that for every s ∈ [ n − 2 , n ] s \in [n - 2,n] there exists a homogeneously embedded wild Cantor set C s {C^s} in R n , n ≥ 3 \mathbb {R}^n, n \geq 3 , of (local) Hausdorff dimension s s . Also, it is shown that for every s ∈ [ n − 2 , n ] s \in [n - 2,n] and for any integer k ≠ n k \ne n such that 1 ≤ k ≤ s 1 \leq k \leq s , there exist everywhere wild k k -spheres and k k -cells, in R n , n ≥ 3 \mathbb {R}^n, n \geq 3 , of (local) Hausdorff dimension s s .

In this paper, we show that for a separable metric space X, every continuous function from a subset S of X into a finite discrete space extends to a continuous function on X if and only if every continuous function from S into any discrete space extends to a continuous function on X. We also show that if there is no inner model having a measurable cardinal, then there is a metric space X with a subspace S such that every 2-valued continuous function from S extends to a continuous function on all of X, but not every discrete-valued continuous function on S extends to such a map on X. Furthermore, if Martin's Axiom is assumed, such a space can be constructed so that not even co-valued functions on S need extend. This last result uses a version of the Isbell-Mrowka space 'P having a C· -embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such 'P exists.

The author constructs a locally connected, homogeneous, finitedimensional, compact, metric space which is not bihomogeneous, thus providing a compact counterexample to a problem posed by B. Knaster around 1921.

We introduce the Leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological Conley index of an isolated invariant set of a homeomorphism on a locally compact metric space. We prove the homotopy and additivity properties for this index and compute the index in some examples. As one of applications we prove the existence of nonconstant, bounded solutions of the Euler approximation of a certain system of ordinary differential equations.

Topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy. Topological entropy for one parameter flows on a compact metric spaces is defined by Bowen. General statements are proved about this entropy, but it is not easy to calculate the topological entropy, and to show it is invariant under conjugacy. For all this I would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits. Some well-known results are proved as well using this definition. These results enable us to prove some results which seem difficult to prove using Bowen’s definition. Also we show here that this definition is equivalent to Bowen’s definition for any flow without fixed points on a compact metric space. Finally, it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections.

R \mathbf {R} -trees arise naturally in the study of groups of isometries of hyperbolic space. An R \mathbf {R} -tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals R \mathbf {R} . Actions on R \mathbf {R} -trees can be viewed as ideal points in the compactification of groups of isometries. As such they have applications to the study of hyperbolic manifolds. Our concern in this paper, however, is with the topological characterization of R \mathbf {R} -trees. Our main theorem is the following: Let ( X , p ) (X,p) be a metric space. Then X X is uniquely arcwise connected and locally arcwise connected if, and only if, X X admits a compatible metric d d such that ( X , d ) (X,d) is an R \mathbf {R} -tree. Essentially, we show how to put a convex metric on a uniquely arcwise connected, locally arcwise connected, metrizable space.

Applying ideas from topological dynamics in compact metric spaces to the Stone-Cěch compactification of a discrete semigroup, several new proofs of old results and some new results in Ramsey Theory are obtained. In particular, two ultrafilter proofs of van der Waerden’s Theorem are given. An ultrafilter approach to "central" sets (sets which are combinatorially rich) is developed. This enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central. Also, a fortuitous answer to a question of Ellis is obtained.

Quantum isometry groups of spectral triples associated with ap- proximately finite-dimensional C � -algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli dia- grams. This is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middle- third Cantor set. It is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural Laplacians.

We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into ℝ d , where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n 1/12 is NP-hard. From this result we derive inapproximability with a factor roughly n 1/(22d-10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Vaisala), For d > 3, we obtain a stronger inapproximability result by a different reduction: assuming P≠NP, no polynomial-time algorithm can distinguish between spaces embeddable in ℝ d with constant distortion from spaces requiring distortion at least n c/d , for a constant c > 0. The exponent c/d has the correct order of magnitude, since every n-point metric space can be embedded in ℝ d with distortion O(n 2/d log 3/2 n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in ℝ 2 , while all not too large subspaces of it embed almost isometrically.