# types of functions between metric spaces

Let $X,Y$ be metric spaces (by metric space I mean $(X,d)$ with $d:X\times X\to [0,\infty ]$ satisfying the triangle inequality, symmetry and $d(x,x)=0$). The following definitions for a function $f:X\to Y$ are well-known:

1) $f$ is nonexpanding if $d(f(x),f(x'))\le d(x,x')$.

2) $f$ is Lipschitz if there exist a constant $C>0$ such that $d(f(x),f(x')) < C\cdot d(x,x')$.

3) $f$ is uniformly continuous if for all $\epsilon >0$ there is $\delta >0$ such that if $d(x,x')< \delta$ then $d(f(x),f(x'))< \epsilon$.

4) $f$ is continuous if for all $x\in X$ and $\epsilon > 0$ there is $\delta >0$ such that if $d(x,x')< \delta$ then $d(f(x),f(x'))< \epsilon$.

5) $f$ is large scale Lipschitz if there exist constants $C,A$ such that $d(f(x),f(x')) < C\cdot d(x,x') + A$.

6) $f$ is bornologous if for all $R>0$ there is $S>0$ such that if $d(x,x')<R$ then $d(f(x),f(x')) < S$.

7) $f$ is weak bornologous if there exist $R,S>0$ such that if $d(x,x')<R$ then $d(f(x),f(x')) < S$.

Clearly $1\implies 2 \implies 3 \implies 4$ and $5\implies 6 \implies 7$.

A definition I did not find written anywhere is the following. First an auxiliary concept: a function $\alpha : [0,\infty ]\to [0,\infty ]$ satisfying $\alpha (0)=0$, $\alpha$ is nondecreasing, $\alpha (s+t)\le \alpha (s)+\alpha (t)$, and for all $\epsilon >0$ there is $\delta >0$ such that $\alpha (\delta )<\epsilon$ shall be called a distance distortion function. Any distance distortion function associates with any metric space $X$ a new metric space, $\alpha _*(X)$, with the same underlying set, and with distance function $\alpha_*(d):X\times X\to [0,\infty ]$, given by $\alpha _*(X)(x,x')=\alpha (d(x,x'))$.

2.5) say that $f$ is relatively nonexpanding if there exists a distance distortion function $\alpha$ such that $f:\alpha _*(X)\to Y$ is nonexpanding.

It is clear that $2\implies 2.5 \implies 3$.

So my questions are:

• Does the concept given by definition 2.5 have a name, if so, any references will be greatly appreciated.

• Is the list of concepts above a complete list of notions for structure preserving functions between metric spaces, for sensible notions of structure (e.g., above, rigid metric preservation, weaker metric preservations, uniform preservation, continuity, and the large scale notions).

Thanks!

Later addition: Condition 2.5 is known as a function admitting a special modulus of continuity, in particular subadditive/concave (thanks @5pm for that information, thus answering my first question).

• 2.5) says that $f$ admits a global modulus of continuity. It is equivalent to having a pointwise finite function $\alpha$ (with limit $0$ at $0$) such that $d_Y(f(a),f(b))\le \alpha(d_X(a,b))$ for all $a,b$. The concavity and monotonicity can be arranged by taking the concave majorant. – user53153 Mar 9 '13 at 5:00
• thaks @5pm. Any good reference you can recommend or shall I just start googling? – Ittay Weiss Mar 9 '13 at 5:04
• The Wikipedia article on modulus of continuity calls them "special uniformly continuous functions". Yes, googling is probably the way to go. – user53153 Mar 9 '13 at 5:14
• Yes, I see. Thanks again. – Ittay Weiss Mar 9 '13 at 5:19

## 1 Answer

More remarks about (2.5):

• $(2.5) \iff (3) + (6)$. That is, (2.5) describes bornologous uniformly continuous maps.
• A comment on "special" moduli of continuity. The only really special property is being finite. As long as there is a finite modulus of continuity, we can take the concave majorant of it, which is also finite.
• (2.5) can be rephrased once more as: for every $\epsilon>0$ there exists $\delta>0$ such that (... as in uniform continuity), and additionally the supremum of allowed $\delta$ tends to infinity as $\epsilon\to \infty$. This form has some shock value: $\epsilon\to\infty$.

For the second question. There are also "two-sided" notions that require invertibility and having the inverse with the same property. Isometries, bi-Lipschitz maps, and uniform homeomorphisms are examples of these. Then there are quasisymmetric maps (which do not admit a description by a pair of one-sided conditions); even more general quasi-Möbius maps, quasi-isometries (not to be confused with bi-Lipschitz maps), and maps preserving other coarse structures. And probably a few more that were invented by someone as I was typing this.

• good comments. Just to clarify on the second question, the two-sided notions is not what I'm looking for. These are just the isomorphisms in the appropriate categories of metric spaces and whatever choice of structure preserving maps. I'm not sure where to place quasimetric maps, I need to think about that a bit. I'm not sure why you place 'coarse structures' in the plural. Coarse maps is certainly a missing concept in my list, but what other coarse structures do you mean? – Ittay Weiss Mar 9 '13 at 8:55
• @IttayWeiss The Wikipedia article on coarse structures (to which I linked) lists more than one. – user53153 Mar 9 '13 at 15:07
• not quite. One definition. Several examples. – Ittay Weiss Mar 9 '13 at 20:13
• @IttayWeiss Every example of a coarse structure leads to a definition of a type of maps, namely those preserving the particular structure. – user53153 Mar 9 '13 at 20:19
• only the first example, the bounded structure, starts with a metric space to produce a coarse structure. The other examples are abstract, so don't correspond to metric notions. – Ittay Weiss Mar 9 '13 at 20:31