Nonexpanding Maps Research Articles

On $\ast$-measure monads on the category of ultrametric spaces

The functor of $\ast$-measures of compact support on the category of ultrametric spaces and non-expanding maps is introduced in the previous publication of the authors. In the present note, we prove that this functor determines a monad on this category. The monad structure allows us to define the tensor product of $\ast$-measures. We consider some applications of this notion to equilibrium theory.

The horofunction boundary of a Gromov hyperbolic space

We highlight a condition, the approaching geodesics property, on a proper geodesic Gromov hyperbolic metric space, which implies that the horofunction compactification is topologically equivalent to the Gromov compactification. It is known that this equivalence does not hold in general. We prove using rescaling techniques that the approaching geodesics property is satisfied by bounded strongly pseudoconvex domains of $$\mathbb {C}^q$$ endowed with the Kobayashi metric. We also show that bounded convex domains of $$\mathbb {C}^q$$ with boundary of finite type in the sense of D’Angelo satisfy a weaker property, which still implies the equivalence of the two said compactifications. As a consequence we prove that on those domains big and small horospheres as defined by Abate in (Math Z 198(2):225–238, 1988) coincide. Finally we generalize the classical Julia’s lemma, giving applications to the dynamics of non-expanding maps.

On spaces of *-measures on ultrametric spaces

The notion of * -measure on a compact Hausdorff space is introduced and investigated in a previous publication of the first named author. In the present note we consider the set of all * -measures of compact support on an ultrametric space. An ultrametrization of this set is provided, which determines a functor in the category of ultrametric spaces and non-expanding maps. We prove that this functor is locally non-expanding and preserves the class of complete ultrametric spaces.

Fuzzy metrization of the spaces of idempotent measures

In idempotent mathematics, the idempotent measures (Maslov measures) are counterparts of the probability measures. We provide a fuzzy metrization of the set of idempotent measures on fuzzy metric spaces. We prove that this fuzzy metrization determines a monad in the category of fuzzy metric spaces and non-expanding maps.

Metrization of free groups on ultrametric spaces

We consider ultrametrizations of free topological groups of ultrametric spaces. A construction is defined that determines a functor in the category UMET 1 of ultrametric spaces of diameter ⩽1 and nonexpanding maps. This functor is the functorial part of a monad in UMET 1 and we provide a characterization of the category of its algebras.

Volumes and areas of Lipschitz metrics

We develop and generalize methods of estimating (Riemannian and Finsler) filling volumes using nonexpanding maps to Banach spaces of L∞ type. For every Finsler volume functional (such as the Busemann volume or the Holmes–Thompson volume) we construct a natural extension from the class of Finsler metrics to all Lipschitz metrics and define the notion of area for Lipschitz surfaces in a Banach space. We establish a correspondence between minimal fillings and minimal surfaces in L∞ type spaces. We introduce a Finsler volume functional for which Riemannian and Finsler filling volumes are equal and prove that this functional is semi-elliptic.

Probability measure monad on the category of ultrametric spaces

The set of all probability measures with compact support on an ultrametric space can be endowed with a natural ultrametric. We show that the functor of probability measures with finite supports (respectively compact supports) forms a monad in the category of ultrametric spaces (respectively complete ultrametric spaces) and nonexpanding maps. It is also proven that the G-symmetric power functor has an extension onto the Kleisli category of the probability measure monad.

Idempotent probability measures on ultrametric spaces

Following the construction due to Hartog and Vink we introduce a metric on the set of idempotent probability measures (Maslov measures) defined on an ultrametric space. This construction determines a functor on the category of ultrametric spaces and nonexpanding maps. We prove that this functor is the functorial part of a monad on this category. This monad turns out to contain the hyperspace monad.

Maps between a space and its square

We construct metric (or uniform or topological) spaces X and Y such that every non-expanding (or uniformly continuous or continuous) map Y× Y→ Y depends only on one of its variables, the space X fails to have this property, and the respective monoids of all non-expanding (or uniformly continuous or continuous) selfmaps of X and Y are the same.

The category of ultrametric spaces is isomorphic to the category of complete, atomic, tree-like, and real graduated lattices LAT*

It is proved that for any ultrametric space (X, d), the set L(X) of its closed balls is a lattice $$ (\mathbf{L}(X), \bigcap, \mathrm{sup}, r(B)) $$ . It is complete, atomic, tree-like, and real graduated. For any such lattice $$ (L, \bigwedge, \bigvee, r) $$ , the set A(L) of its atoms can be naturally equipped with an ultrametric $$ \Delta (x,y) $$ . These assignments are inverse of one another: $$ (\mathbf{A}(\mathbf{L}(X)), \Delta) = (X,d)\quad \mathrm{and}\quad (L, \bigwedge, \bigvee, r) = (\mathbf{L}(\mathbf{A}(L)), \bigcap, \mathrm{sup}, r(B)) $$ where the first equality means an isometry while the second one is a lattice isomorphism. A similar correspondence established for morphisms, shows that there is an isomorphism of categories. The category ULTRAMETR of ultrametric spaces and non-expanding maps is isomorphic to the category LAT* of complete, atomic, tree-like, real graduated lattices and isotonic, semi-continuous, non-extensive maps. We describe properties of the isomorphism functor and its relations to the categorical operations and action of other functors. Basic properties of a space (such as completeness, spherical completeness, total boundedness, compactness, etc.) are translated into algebraic properties of the corresponding lattice L(X).

The structure of equicontinuous maps

Let $(X,d)$ be a metric space, and $f:X\rightarrow X$ be a continuous map. In this paper we prove that if $R(f)$ is compact, and $\omega (x,f)\not =\emptyset$ for all $x\in X$, 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 $(h\times g)|S$ of the product map $h\times g$. In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.

On ultrametrization of general metric spaces

This paper gives a complete description of ultrametric spaces up to uniform equivalence. It also describes all metric spaces which can be mapped onto ultrametric spaces by a non-expanding one-to-one map. Moreover, it describes particular classes of spaces, for which such a map has a continuous (uniformly continuous) inverse map. This gives a complete solution for the Hausdorff-Bayod Problem (what metric spaces admit a subdominant ultrametric?) as well as for two other problems posed by Bayod and Martinez-Maurica in 1990. Further, we prove that for any metric space (X,d), there exists the greatest non-expanding ultrametric image of X (an ultrametrization of X), i.e., the category of ultrametric spaces and non-expanding maps is a reflective subcategory in the category of all metric spaces and the same maps. In Section II, for any cardinal r, we define a complete ultrametric space L τ of weight τ such that any metric space X of weight r is an image of a subset L(X) of L τ under a non-expanding, open, and compact map with totally-bounded pre-irnages of compact subsets. This strengthens Hausdorff-Morita, Morita-de Groot and Nagami theorems. We also construct an ultrametric space L(τ), which is a universal pre-image of all metric spaces of weight τ under non-expanding open maps. We define a functor A from the category of ultrametric spaces to a category of Boolean algebras such that algebras A(X) and A(Y) are isomorphic iff the completions of spaces X and Y are uniformly homeomorphic. Some properties of the functor A and the ultrametrization functor are discussed.

Non-expanding maps and Busemann functions

We give stronger versions and alternative simple proofs of some results of Beardon, [Be1] and [Be2]. These results concern contractions of locally compact metric spaces and generalize the theorems of Wolff and Denjoy about the iteration of a holomorphic map of the unit disk. In the case of unbounded orbits, there are two types of statements which can sometimes be proven; first, about invariant horoballs, and second, about the convergence of the iterates to a point on the boundary. A few further remarks of similar type are made concerning certain random products of semicontractions and also concerning semicontractions of Gromov hyperbolic spaces.

Representation of algebraic theories and non-expanding maps

Finite product preserving full faithful functors of monosorted finitary algebraic theories into categories with finite products are investigated. One consequence of a general method presented here is the result that for every triple m≤n≤p of natural numbers, there exist metric spaces X 1=(P,d 1) and X 2=(P,d 2) such that the categories N k(X i) formed by all non-expanding maps between finite powers X i 0 , X i 1 , …, X i k−1 of X i with i=1,2 have the following properties: N k(X 1)is formed by the same maps as N k(X 2) iff k≤m, N k(X 1)is isomorphic to N k(X 2) iff k≤n, N k(X 1) is elementarily equivalent to N k(X 2) iff k≤p .

A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces

We study integrable cocycles u.n; x/ over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y , e.g. a Cartan-Hadamard space or a uniformly convex Banach space. It is proved that for any y 2 Y and almost all x; there exist A 0 and a unique geodesic ray .t; x/in Y starting at y such that lim n!1 1 n d. .An; x/; u.n; x/y/ D 0: In the case where Y is the symmetric space GLN.R/=ON.R/ and the cocycles take values in GLN.R/; this is equivalent to the multiplicative ergodic theorem of Oseledec. Two applications are also described. The first concerns the determination of Poisson boundaries and the second concerns Hilbert-Schmidt operators.

A fixed box theorem for the cartesian product of graphs and metric spaces

We prove the following result: Let ( X, δ) = ∏ i∈ I ( X i , δ i ), be a product of a finite number of finite metric spaces, where the distance in X is the sum of the coordinate distances. Then for each non-expanding map ϕ : X → X, there exists a set R ⊆ X such that R coincides with the product of its projections and ϕ( R) = R. This extends known results on cubes and Hamming graphs.

The Pseudo-Orbit Shadowing Property for Markov Operators in the Space of Probability Density Functions

AbstractLet X be a space with two metrics d1 and d2. Let S :(X, d1) → (X, d2) be continuous. We say S has the generalized pseudoorbit shadowing property with respect to the metrics d1 and d2 if for every every δ-pseudo-orbit in d1 can be ∊-shadowed by a true orbit in d2, i.e., if {x0, x1,…} satisfies for all i ≧ 0, then for all i ≧ 0. The main result of this note shows that certain Markov operators P : L1→ L1 have the generalized shadowing property on weakly compact subsets of the space of probability density functions, where d1 is the metric of norm convergence and d2 is the metric of weak convergence. An important class of such operators are the Frobenius-Perron operators induced by certain expanding and nonexpanding maps on the interval. When there is exponential convergence of the iterates to the density, we can express δ in terms of ∊. We also show that, unlike the situation in the space X itself, the generalized shadowing property is valid for all parameters in families of maps and that there is stability of the shadowing property.

Constructive approximations to densities invariant under nonexpanding transformations

Let τ: [0, 1]→[0, 1] be a piecewise monotonic, nonexpanding map which has an invariant densityg and is topologically conjugate to a piecewise monotonic, expanding map, where the conjugacy is absolutely continuous. An effective, computable method is presented for approximatingg.

A matrix method for estimating the Liapunov exponent of one-dimensional systems

Let τ: [0, 1]→[0, 1] be a piecewise monotonie expanding map. Then τ admits an absolutely continuous invariant measureμ. A result of Kosyakin and Sandler shows thatμ can be approximated by a sequence of absolutely continuous measuresμn invariant under piecewise linear Markov maps τitn. Each τitn is constructed on the inverse images of the turning points of τ. The easily computable measuresμn are used to estimate the Liapunov exponent of τ. The idea of using Markov maps for estimating the Liapunov exponent is applied to both expanding and nonexpanding maps.

Approximating the σ-finite measure invariant under a non-expanding map

Let τ be a non-expanding map which admits a σ-finite, absolutely continuous invariant measure μ. The measure μ¦ E is approximated as closely as desired by a measure μ n invariant under the transformation τ n satisfying inf x ¦τ′ n(x)¦ > 1 . The orbit { τ n j ( x)} j = 0 ∞ “exhibits” μ¦ E much more quickly than does the orbit { τ j ( x)} j = 0 ∞.