Asymptotic behavior of metric spaces at infinity

TL;DR

This paper introduces a new sequential method to analyze the structure of metric spaces at infinity, providing criteria for their finiteness and boundedness.

Contribution

It proposes a novel sequential approach to study the asymptotic structure of metric spaces and establishes criteria for their finiteness and boundedness at infinity.

Findings

Criteria for finiteness at infinity

Criteria for boundedness at infinity

Introduction of a sequential analysis approach

Abstract

A new sequential approach to investigations of structure of metric spaces at infinity is proposed. Criteria for finiteness and boundedness of metric spaces at infinity are found.

Equations

$$(\tilde{x}\equiv \tilde{y})\iff \left(\underset{n\to \infty }{\lim }\frac{d(x_n,y_n)}{r_n}=0\right).$$

$$\rho (\alpha ,\beta ):=\underset{n\to \infty }{\lim }\frac{d(x_n,y_n)}{r_n}.$$

$$\left(\{u,v\}\in E(G_{X,\tilde{r}})\right)\iff \left(u\text{mbox}andv\text{mbox}aremutuallystableandu\ne v\right),$$

$$\left(\tilde{z}\in {\alpha }_0\right)\iff \left(\tilde{z}\in {\tilde{\mathbf{X}}}_{\infty ,\tilde{r}}\text{ and }{\widetilde{\stackrel{~}{d}}}_{\tilde{r}}(\tilde{z})=0\right),$$

$$\underset{k\to \infty }{\lim }\frac{d(x_{n_k},y_{n_k})}{r_{n_k}}=\underset{n\to \infty }{\lim }\frac{d(x_n,y_n)}{r_n}.$$

$$\begin{array}{ccc}{\tilde{\mathbf{X}}}_{\infty ,\tilde{r}}& \stackrel{{\phi }_{{\tilde{r}}^{\prime }}}{\to }& {\tilde{\mathbf{X}}}_{\infty ,{\tilde{r}}^{\prime }}\\ \pi \downarrow & & \downarrow {\pi }^{\prime }\\ {\mathbf{\Omega }}_{\infty ,\tilde{r}}^X& \stackrel{e{m}^{\prime }}{\to }& {\mathbf{\Omega }}_{\infty ,{\tilde{r}}^{\prime }}^X\end{array}$$

$$F_{n}(x_{1},...,x_{n}):=\begin{cases}\frac{\mathop{\min}\limits_{1\leq k\leq n}d(x_{k},p)\mathop{\prod}\limits_{1\leq k<l\leq n}d(x_{k},x_{l})}{\left(\mathop{\max}\limits_{1\leq k\leq n}d(x_{k},p)\right)^{\frac{n(n-1)}{2}+1}},&\mbox{if}$ $(x_{1},...,x_{n})\neq(p,...,p)\\ 0,&\mbox{if}$ $(x_{1},...,x_{n})=(p,...,p).\\ \end{cases}$$

$$p(E,\infty ):=\underset{h\to \infty }{\mathrm{limsup}}\frac{l(\infty ,h,E)}{h}$$

$$\underset{n\to \infty }{\lim }\frac{d(x_n,p)}{r_n}=1.$$

$$\underset{Y\in \mathcal{F}}{\sup }\mathrm{diam}Y<\infty .$$

$$\underset{Y\in \mathcal{F}}{\inf }\inf \{d_Y(p_Y,y):y\in Y\setminus \{p_Y\}\}>0,$$

$$\text{Lim}\underset{n\to \infty }{\inf }{K}_n:=\{y\in Y:\underset{n\to \infty }{\lim }\text{dist}(y,K_n)=0\}$$

$${\Omega }_{\infty ,({\tilde{r}}_1,...,{\tilde{r}}_m)}^X:={\Omega }_{\infty ,{\tilde{r}}_m}^{{\Omega }_{\infty ,(r_1,...,r_{m-1})}^X}$$

Full text

UDK 515.124

MATHEMATICS

Institute of Applied Mathematics and Mechanics NAS of Ukraine, Dobrovolskogo Str. 1, Sloviansk, 84100, Ukraine; Phone: +38(0626)66 55 00

Emails: viktoriiabilet@gmail.com, oleksiy.dovgoshey@gmail.com

Asymptotic behavior of metric spaces at infinity

Viktoriia Bilet, Oleksiy Dovgoshey

A new sequential approach to investigations of structure of metric spaces at infinity is proposed. Criteria for finiteness and boundedness of metric spaces at infinity are found.

Keywords: asymptotic boundedness of metric space, asymptotic finiteness of metric space, convergence of metric spaces, strong porosity at a point.

1 Introduction

Under the asymptotic structure describing the behavior of an unbounded metric space $(X,d)$ at infinity we mean a metric space that is a limit of rescaling metric spaces $(X,\frac{1}{r_{n}}d)$ for $r_{n}$ tending to infinity. The Gromov–Hausdorff convergence and the asymptotic cones are most often used for construction of such limits. Both of these approaches are based on higher-order logic abstractions (see, for example, [1] for details), which makes them very powerful, but it does away the constructiveness. In this paper we propose a more elementary, sequential approach for describing the structure of unbounded metric spaces at infinity.

Let $(X,d)$ be an unbounded metric space, $p$ be a point of $X$ and $\tilde{r}=(r_{n})_{n\in\mathbb{N}}$ be a scaling sequence of positive real numbers tending to infinity. Denote by $\mathbf{\tilde{X}}_{\infty,\tilde{r}}$ the set of all sequences $\tilde{x}=(x_{n})_{n\in\mathbb{N}}\subset X$ for each of which $\mathop{\lim}\limits_{n\to\infty}d(x_{n},p)=\infty$ and there is a finite limit $\tilde{\tilde{d}}_{\tilde{r}}(\tilde{x}):=\mathop{\lim}\limits_{n\to\infty}\frac{d(x_{n},p)}{r_{n}}$. Define the equivalence relation $\equiv$ on $\mathbf{\tilde{X}}_{\infty,\tilde{r}}$ as

[TABLE]

Let $\mathbf{\Omega}_{\infty,\tilde{r}}^{X}$ be the set of equivalence classes generated by $\equiv$ on $\mathbf{\tilde{X}}_{\infty,\tilde{r}}$. We shall say that points $\alpha,\beta\in\mathbf{\Omega}_{\infty,\tilde{r}}^{X}$ are mutually stable if for $\tilde{x}\in\alpha$ and $\tilde{y}\in\beta$ there is a limit

[TABLE]

Let us consider the weighted graph $(G_{X,\tilde{r}},\rho)$ with the vertex set $V(G_{X,\tilde{r}})=\mathbf{\Omega}_{\infty,\tilde{r}}^{X}$, and the edge set $E(G_{X,\tilde{r}})$ such that

[TABLE]

and the weight $\rho:E(G_{X,\tilde{r}})\to\mathbb{R}^{+}$ defined by formula (1.1).

Definition 1.1**.**

The pretangent spaces (to $(X,d)$ at infinity w.r.t. $\tilde{r}$) are the maximal cliques $\Omega_{\infty,\tilde{r}}^{X}$ of $G_{X,\tilde{r}}$ with metrics determined with the help of (1.1).

Recall that a clique in a graph $G$ is a set $A\subseteq V(G)$ such that every two distinct points of $A$ are adjacent. A clique $C$ in $G$ is maximal if $C\subseteq A$ implies $C=A$ for every clique $A$ in $G$.

Define the subset $\alpha_{0}=\alpha_{0}(X,\tilde{r})$ of the set $\tilde{\mathbf{X}}_{\infty,\tilde{r}}$ by the rule:

[TABLE]

Then $\alpha_{0}$ is a common point of all pretangent spaces $\Omega_{\infty,\tilde{r}}^{X}$. It means in particular that the graph $G_{X,\tilde{r}}$ is connected.

Let $(n_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$ be infinite and strictly increasing. Denote by $\tilde{r}^{\prime}$ the subsequence $(r_{n_{k}})_{k\in\mathbb{N}}$ of the $\tilde{r}=(r_{n})_{n\in\mathbb{N}}$ and, for every $\tilde{x}=(x_{n})_{n\in\mathbb{N}}\in\mathbf{\tilde{X}}_{\infty,\tilde{r}}$, write $\tilde{x}^{\prime}:=(x_{n_{k}})_{k\in\mathbb{N}}$. It is clear that $\mathop{\lim}\limits_{k\to\infty}d(x_{n_{k}},p)=\infty$ and $\tilde{\tilde{d}}_{\tilde{r}^{\prime}}(\tilde{x}^{\prime})=\tilde{\tilde{d}}_{\tilde{r}}(\tilde{x})$ for every $\tilde{x}\in\mathbf{\tilde{X}}_{\infty,\tilde{r}}$. Moreover, if $\tilde{y}\in\mathbf{\tilde{X}}_{\infty,\tilde{r}}$ and $\mathop{\lim}\limits_{n\to\infty}\frac{d(x_{n},y_{n})}{r_{n}}$ exists, then

[TABLE]

Let us define $\tilde{\mathbf{X}}_{\infty,\tilde{r}^{\prime}}$ and $\rho^{\prime}$ similarly to $\tilde{\mathbf{X}}_{\infty,\tilde{r}}$ and, respectively, $\rho$ and let $\pi\colon\tilde{\mathbf{X}}_{\infty,\tilde{r}}\to\mathbf{\Omega}_{\infty,\tilde{r}}$, $\pi^{\prime}\colon\tilde{\mathbf{X}}_{\infty,\tilde{r}}\to\mathbf{\Omega}_{\infty,\tilde{r}^{\prime}}$ be the natural projections $\pi(x):=\{\tilde{y}\in\tilde{\mathbf{X}}_{\infty,\tilde{r}}\colon\rho(\tilde{x},\tilde{y})=0\}$, $\pi^{\prime}(x):=\{\tilde{y}\in\tilde{\mathbf{X}}_{\infty,\tilde{r}^{\prime}}\colon\rho^{\prime}(\tilde{x},\tilde{y})=0\}$ and let $\varphi_{\tilde{r}^{\prime}}(\tilde{x}):=\tilde{x}^{\prime}$ for all $\tilde{x}\in\tilde{\mathbf{X}}_{\infty,\tilde{r}}$. Then there is an embedding $em^{\prime}\colon\mathbf{\Omega}_{\infty,\tilde{r}}^{X}\to\mathbf{\Omega}_{\infty,\tilde{r}}^{X}$ of the weighted graph $(G_{X,\tilde{r}},\rho)$ in the weighted graph $(G_{X,\tilde{r}^{\prime}},\rho^{\prime})$ such that the diagram

[TABLE]

is commutative. Since $em^{\prime}$ is an embedding of weighted graphs, $em^{\prime}(C)$ is a clique in $G_{X,\tilde{r}^{\prime}}$ if $C$ is a clique in $G_{X,\tilde{r}}$. Furthermore, (1.3) implies that the restrictions $em^{\prime}|_{\Omega_{\infty,\tilde{r}}^{X}}$ are isometries of the pretangent spaces $\Omega_{\infty,\tilde{r}}^{X}$ on the metric spaces $em^{\prime}(\Omega_{\infty,\tilde{r}}^{X})$.

Definition 1.2**.**

A pretangent space $\Omega_{\infty,\tilde{r}}^{X}$ is tangent if the clique $em^{\prime}(\Omega_{\infty,\tilde{r}}^{X})$ is maximal for every infinite, strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$.

Example 1.3**.**

Let $E$ be a finite-dimensional Euclidean space and let $X\subseteq E$ such that the Hausdorff distance $d_{H}(E,X)$ is finite. Then for every scaling sequence $\tilde{r}$ all pretangent spaces $\Omega_{\infty,\tilde{r}}^{X}$ are tangent and isometric to $E$.

In conclusion of this brief introduction it should be noted that there exist other techniques which allow to investigate the asymptotic properties of metric spaces at infinity. As examples, we mention only the balleans theory [2] and the Wijsman convergence [3], [4], [5].

2 Finiteness

In this section we study the conditions under which pretangent spaces are finite.

Theorem 2.1**.**

Let $(X,d)$ be an unbounded metric space, $p\in X$, $n\geq 2$ and let

[TABLE]

Then the inequality $\left|\Omega_{\infty,\tilde{r}}^{X}\right|\leq n$ holds for every pretangent space $\Omega_{\infty,\tilde{r}}^{X}$ if and only if $\mathop{\lim}\limits_{x_{1},...,x_{n}\to\infty}F_{n}(x_{1},...,x_{n})=0.$

Note that, for every unbounded metric space $(X,d)$, there is a pretangent space $\Omega_{\infty,\tilde{r}}^{X}$ consisting at least two points.

Corollary 2.2**.**

Let $(X,d)$ be an unbounded metric space and let $\alpha_{0}$ be a point defined by (1.2) for every scaling sequence $\tilde{r}.$ Then the following statements are equivalent.

1. (i)

The graph $G_{X,\tilde{r}}$ is a star with the center $\alpha_{0}$ for every scaling sequence $\tilde{r}$; 2. (ii)

The limit relation $\mathop{\lim}\limits_{x_{1},x_{2}\to\infty}F_{2}(x_{1},x_{2})=0$ holds.

Let us consider now the problem of existence of finite tangent spaces.

Definition 2.3**.**

Let $E\subseteq\mathbb{R}^{+}$. The porosity of $E$ at infinity is the quantity

[TABLE]

where $l(\infty,h,E)$ is the length of the longest interval in the set $[0,h]\setminus E.$ The set $E$ is strongly porous at infinity if $p(E,\infty)=1$.

The standard definition of the porosity at a point can be found in [6].

For a metric space $(X,d)$ and $p\in X$ write $S_{p}(X):=\{d(x,p)\colon x\in X\}$.

Theorem 2.4**.**

Let $(X,d)$ be an unbounded metric space, $p\in X.$ The following statements are equivalent:

1. (i)

The set $S_{p}(X)$ is strongly porous at infinity; 2. (ii)

There is a single-point tangent space $\Omega_{\infty,\tilde{r}}^{X}$; 3. (iii)

There is a finite tangent space $\Omega_{\infty,\tilde{r}}^{X}$; 4. (iv)

There is a compact tangent space $\Omega_{\infty,\tilde{r}}^{X}$; 5. (v)

There is a bounded, separable tangent space $\Omega_{\infty,\tilde{r}}^{X}$.

Some results which are similar to Theorem 2.1 and Theorem 2.4 can be found in [7] and [8] respectively.

3 Boundedness

Let $\tilde{\tau}=(\tau_{n})_{n\in\mathbb{N}}\subset\mathbb{R}$. We shall say that $\tilde{\tau}$ is eventually increasing if the inequality $\tau_{n+1}\geq\tau_{n}$ holds for sufficiently large $n$. For $E\subseteq\mathbb{R}^{+}$ write $\tilde{E}_{\infty}^{i}$ for the set of eventually increasing sequences $\tilde{\tau}\subset E$ with $\mathop{\lim}\limits_{n\to\infty}\tau_{n}=\infty$. Denote also by $\tilde{I}_{E}^{i}$ the set of all sequences of open intervals $(a_{n},b_{n})\subseteq\mathbb{R}^{+}$ meeting the following conditions:

$\bullet$ Each $(a_{n},b_{n})$ is a connected component of the set $\mathop{Int}(\mathbb{R}^{+}\setminus E);$

$\bullet$ $(a_{n})_{n\in\mathbb{N}}$ is eventually increasing;

$\bullet$ $\mathop{\lim}\limits_{n\to\infty}a_{n}=\infty$ and $\mathop{\lim}\limits_{n\to\infty}\frac{b_{n}-a_{n}}{b_{n}}=1$.

Define an equivalence $\asymp$ on the set of sequences of strictly positive numbers as follows. Let $\tilde{a}=(a_{n})_{n\in\mathbb{N}}$ and $\tilde{\gamma}=(\gamma_{n})_{n\in\mathbb{N}}$. Then $\tilde{a}\asymp\tilde{\gamma}$ if there are some constants $c_{1},c_{2}>0$ such that $c_{1}a_{n}<\gamma_{n}<c_{2}a_{n}$ for every $n\in\mathbb{N}$.

Definition 3.1**.**

Let $E\subseteq\mathbb{R}^{+}$ and let $\tilde{\tau}\in\tilde{E}_{\infty}^{i}$. The set $E$ is $\tilde{\tau}$-strongly porous at infinity if there is a sequence $((a_{n},b_{n}))_{n\in\mathbb{N}}\in\tilde{I}_{E}^{i}$ such that $\tilde{\tau}\asymp\tilde{a}$ where $\tilde{a}=(a_{n})_{n\in\mathbb{N}}$. The set $E$ is completely strongly porous at infinity if $E$ is $\tilde{\tau}$-strongly porous at infinity for every $\tilde{\tau}\in\tilde{E}_{\infty}^{i}$.

Note that every completely strongly porous at infinity set is strongly porous at infinity but not conversely.

Definition 3.2**.**

Let $(X,d)$ be an unbounded metric space and let $p\in X$. A scaling sequence $\tilde{r}$ is normal if $\tilde{r}$ is eventually increasing and there is $\tilde{x}\in\tilde{\mathbf{X}}_{\infty,\tilde{r}}$ such that

[TABLE]

Write $\mathcal{F}_{n}(X)$ for the set of all pretangent spaces $\Omega_{\infty,\tilde{r}}^{X}$ with normal scaling sequences $\tilde{r}$. Under what conditions the family $\mathcal{F}_{n}(X)$ is uniformly bounded?

Recall that a family $\mathcal{F}$ of a metric spaces $(Y,d_{Y})$ is uniformly bounded if

[TABLE]

If all metric spaces $(Y,d_{Y})$ are pointed with marked points $p_{Y}\in Y$ and

[TABLE]

then we say that $\mathcal{F}$ is uniformly discrete (w.r.t. the points $p_{Y}$).

The following theorem is an analog of Theorem 3.11 and Theorem 4.1 from [9].

Theorem 3.3**.**

Let $(X,d)$ be an unbounded metric space and let $p\in X$. Then the following statements are equivalent

1. $(i)$

The family $\mathcal{F}_{n}(X)$ is uniformly bounded. 2. $(ii)$

$S_{p}(X)$* is completely strongly porous at infinity.* 3. $(iii)$

The family $\mathcal{F}_{n}(X)$ is uniformly discrete w.r.t. the points $\alpha_{0}$ defined by (1.2).

If $\mathcal{F}_{n}(X)$ is uniformly bounded, then every pretangent space $\Omega_{\infty,\tilde{r}}^{X}$ is bounded, but the converse, in general, does not hold.

Definition 3.4**.**

The set $E\subseteq\mathbb{R}^{+}$ is $\omega$-strongly porous at infinity if for every sequence $\tilde{\tau}\in\tilde{E}_{\infty}^{i}$ there is a subsequence $\tilde{\tau}^{\prime}$ for which $E$ is $\tilde{\tau}^{\prime}$-strongly porous at infinity.

The following theorem gives a boundedness criterion for pretangent spaces.

Theorem 3.5**.**

Let $(X,d)$ be an unbounded metric space and let $p\in X$. All pretangent spaces to $X$ at infinity are bounded if and only if the set $S_{p}(X)$ is $\omega$-strongly porous at infinity.

The example of $\omega$-strongly porous at infinity set $E\subseteq\mathbb{R}^{+}$ which is not completely strongly porous at infinity can be obtained as a modification of Example 2.10 [10].

Using Theorem 3.5 we can obtain a criterion of existence of an unbounded pretangent space. The condition of such type are importnt for development of a theory of pretangent spaces of the second order or more (i.e., pretangent spaces to pretangent spaces, pretangent spaces to pretangent spaces to pretangent spaces and so on).

Let $(Y,d_{Y})$ be a metric space. Then for every $K\subseteq Y$ and $y\in Y$ we write $\textrm{dist}(y,K)=\inf\{d_{Y}(y,x):x\in K\}.$ The following definition can be found in [11].

Definition 3.6**.**

Let $(K_{n})_{n\in\mathbb{N}}$ be a sequence of subsets of $(Y,d_{Y}).$ The set

[TABLE]

is the Kuratowski lower limit of $(K_{n})_{n\in\mathbb{N}}$ in $(Y,d_{Y}).$

For $A\subseteq\mathbb{R}$ and $t\in\mathbb{R}$ we set $tA:=\{ta:a\in A\}.$

Theorem 3.7**.**

Let $(X,d)$ be an unbounded metric space and let $p\in X.$ Then the following statements are equivalent.

1. $(i)$

There exists an unbounded pretangent space $\Omega_{\infty,\tilde{r}}^{X}.$ 2. $(ii)$

There exists a scaling sequence $\tilde{r}=(r_{n})_{n\in\mathbb{N}}$ such that the Kuratowski lower limit $\mathop{\emph{Lim}\inf}\limits_{n\to\infty}\frac{1}{r_{n}}S_{p}(X)$ is an unbounded subset of $\mathbb{R}^{+}.$ 3. $(iii)$

The set $S_{p}(X)$ is not $w$-strongly porous at infinity.

Let $(\tilde{r}_{m})_{m\in\mathbb{N}}$ be a sequence of scaling sequences. Then for every $m\in\mathbb{N}$ and every unbounded metric space $(X,d)$ we define a pretangent space $\Omega_{\infty,(\tilde{r}_{1},...,\tilde{r}_{m})}^{X}$ by the following inductive rule: $\Omega_{\infty,(\tilde{r}_{1})}^{X}:=\Omega_{\infty,\tilde{r}_{1}}^{X}$ if $m=1$ and

[TABLE]

if $m\geq 2.$

Using Theorem 3.7 we can obtain the following corollary.

Corollary 3.8**.**

Let $(X,d)$ be an unbounded metric space and let $p\in X.$ If the equality $p(S_{p}(X),\infty)=0$ holds, then there is a sequence $(\tilde{r}_{m})_{m\in\mathbb{N}}$ of scaling sequences such that the pretangent space $\Omega_{\infty,(\tilde{r}_{1},...,\tilde{r}_{m})}^{X}$ is unbounded for every $m\in\mathbb{N}.$

Acknowledgments. The research was supported by the grant of the State Fund for Fundamental Research (project F71/20570).