The following proposition shows that if we have a scale stable zoom sequence of mapsρxε as in definitions 7.5 and 7.6 then we can improve every member of the sequence such that all maps from the new zoom sequence have better accuracy near the ”center” of the map x∈X, which justifies the name ”foveal maps”. Definition 7.9 Let ρxε be a scale stable zoom sequence. We define for any
ε∈(0,1)the µ-foveal mapφxε made of all pairs(u, u0)∈B(x, ε)¯ ×B¯(y,1)such that
- ifu∈B(x, εµ)¯ then (u,ρ¯xµ(u0))∈ρxεµ, - or else(u, u0)∈ρxε.
Proposition 7.10 Let ρxε be a scale stable zoom sequence with associated zoom modulus F(·) and scale stability modulusFµ(·). The sequence ofµ-foveal maps
φxε is then a scale stable zoom sequence with zoom modulusF(·) +µFµ(·). More- over, the accuracy of the restricted foveal map φxε ∩ B(x, εµ)¯ ×B¯(y, µ) is bounded by µF(εµ), therefore the right hand side term in the cascading of er- rors inequality (7.3.15), applied for the restricted foveal map, can be improved to2F(εµ).
Proof. Letu∈B(x, εµ). Then there are¯ u0, uε0 ∈B(y, µ) and¯ u”, u”ε∈B(y,¯ 1)
such that (u, u0)∈φxε, (u, u”)∈ρxεµ), (u0, u”)∈ρ¯xµ, (u0ε, u”ε)∈ρxε,µ and
1 µD(u
0, u0
ε) +D(u”, u”ε) ≤ Fµ(ε)
Let u, v ∈ B(x, εµ) and¯ u0, v0 ∈ B(y, µ) such that (u, u¯ 0),(v, v0) ∈ φx ε.
According to the definition of φxε, it follows that there are uniquely defined u”, v” ∈B(y,¯ 1) such that (u, u”),(v, v”)∈ ρxεµ and (u0, u”),(v0, v”) ∈ρ¯xµ. We then have: | 1 εd(u, v)−D(u 0, v0)|= =| 1 εd(u, v)−µD(u”, v”)|= = µ| 1 εµd(u, v)−D(u”, v”)| ≤ µF(εµ) Thus we proved that the accuracy of the restricted foveal map
φxε∩ B(x, εµ)¯ ×B(y, µ)¯
is bounded byµF(εµ): | 1
εd(u, v)−D(u
0, v0)| ≤ µF(εµ) (7.4.17) If u, v ∈ B(x, ε)¯ \ B(x, µ) and (u, u¯ 0),(v, v0) ∈ φxε then (u, u0),(v, v0) ∈ ρxε, therefore
| 1
εd(u, v)−D(u
0, v0)| ≤ F(ε)
Suppose now that (u, u0),(v, v0)∈φxεandu∈B(x, εµ) but¯ v∈B(x, ε)\¯ B(x, µ).¯ We have then: | 1 εd(u, v)−D(u 0, v0)| ≤ ≤ | 1 εd(u, v)−D(u 0 ε, v 0)|+D(u0, u0 ε) ≤ F(ε) +µFµ(ε)
We proved that the sequence ofµ-foveal mapsφxε is a zoom sequence with zoom modulus F(·) +µFµ(·).
In order to prove that the sequence is scale stable, we have to computeφxε,µ, graphically shown in the next figure.
d ε 1 1 µD ρx ε ρx εµ 1 µD 1 εµd ε εµ 1 1 y y D D x µ φx ε φεµ x µ φxε,µ D µ µ ε 2 2
We see that (u0, u”) ∈φxε,µ implies that (u0, u”) ∈ρxε,µ or (u0, u”) ∈ ρxεµ,µ. From here we deduce that the sequence of foveal maps is scale stable and that
ε7→max{Fµ(ε), µFµ(εµ)}
is a scale stability modulus for the foveal sequence.
The improvement of the right hand side for the cascading of errors inequality (7.3.15), applied for the restricted foveal map is then straightforward if we use
(7.4.17).
8
Appendix II: Dilation structures
From definition 7.9 we see that¯
ρxµ◦φxε = ρxεµ (8.0.1) Remark that if theµ-foveal mapφxε coincides with the chart ρxε for everyε (that is, if the zoom sequence ρxε is already so good that it cannot be improved by the construction of foveal maps), then relation (8.0.1) becomes
¯
ρxµ◦φxε = φxεµ (8.0.2)
By proposition 7.7, it follows that µ-foveal map at scale εµ is just a 1/µ dilation of a part of the µ-foveal map at scaleε.
An idealization of these ”perfect”, stable zoom sequences which cannot be improved by the µ-foveal map construction for any µ ∈ (0,1), are dilation structures.
There are several further assumptions, which clearly amount to yet other idealizations. These are the following:
- the ”map is the territory assumption”, namelyY =U(x), the ”map space” is included inX, the ”territory”, andy=x.
- ”functions instead relations”, that is the perfect stable zoom sequences ρxε=φxε are graphs of functions, called dilations. That means:
ρxε ⊂ {(δ x
εu0, u0) : u0 ∈Y =Vε(x)}
- ”hidden uniformity”, that is: in order to pass to the limit in various situations, we could choose the zoom modulus and stability modulus to not depend on x∈ X. This innocuous assumption is the least obvious, but necessary one.
With these idealizations in force, remember that we want our dilations to form a stable zoom sequence and we want also the subtler viewpoint stability, which consists in being able to change the point of view in a coherent way, as the scale goes to zero. These are the axioms of a dilation structure.
We shall use here a slightly particular version of dilation structures. For the general definition of a dilation structure see [2]. More about this, as well as about length dilation structures, see [5].
Definition 8.1 Let (X, d)be a complete metric space such that for any x∈X
the closed ball B(x,¯ 3) is compact. A dilation structure (X, d, δ) over (X, d)is the assignment to anyx∈X andε∈(0,+∞)of a homeomorphism, defined as: if ε∈(0,1]thenδεx:U(x)→Vε(x), elseδεx:Wε(x)→U(x), with the following properties.
A0. For anyx∈X the setsU(x), Vε(x), Wε(x) are open neighbourhoods ofx. There are1< A < B such that for anyx∈X and anyε∈(0,1)we have:
Bd(x, ε)⊂δxεBd(x, A)⊂Vε(x)⊂
⊂Wε−1(x)⊂δxεBd(x, B)
Moreover for any compact set K ⊂ X there are R = R(K) > 0 and
ε0=ε(K)∈(0,1) such that for allu, v∈B¯d(x, R)and allε∈(0, ε0), we haveδxεv∈Wε−1(δxεu).
A1. For anyx∈X δεxx=xand δ1x=id. Consider the closureCl(dom δ) of the set
dom δ={(ε, x, y)∈(0,+∞)×X×X : ifε≤1 theny∈U(x), elsey∈Wε(x)}
seen in[0,+∞)×X ×X endowed with the product topology. The func- tionδ : dom δ →X,δ(ε, x, y) =δxεy is continuous, admits a continuous extension overCl(dom δ)and we have lim
ε→0δ x εy = x.
A2. For any x,∈X, ε, µ∈(0,+∞)and u∈U(x), whenever one of the sides are well defined we have the equalityδxεδµxu=δxεµu.
A3. For any xthere is a distance function (u, v)7→dx(u, v), defined for any
u, v in the closed ball (in distance d) B(x, A)¯ , such that uniformly with respect toxin compact set we have the limit:
lim ε→0 sup | 1 εd(δ x εu, δ x εv) − d x(u, v)| : u, v∈B¯ d(x, A) = 0
A4. Let us define∆xε(u, v) =δδ
x εu
ε−1δ
x
εv. Then we have the limit, uniformly with respect tox, u, v in compact set,
lim ε→0∆ x ε(u, v) = ∆ x (u, v)
8.1
Dilations as morphisms: towards the chora
It is algebraically straightforward to transport a dilation structure: given (X, d, δ) a dilation structure and f : X → Z a uniformly continuous homeomorphism from X (as a topological space) to another topological spaceZ (actually more than a topological space, it should be a space endowed with an uniformity), we can define the transport of (X, d, δ) byf as the dilation structure (Z, f∗d, f∗δ). The distancef ∗dis defined as
(f∗d) (u, v) =d(f(u), f(v))
which is a true distance, because we supposed f to be a homeomorphism. For any u, v ∈ X and ε > 0, we define the new dilation based at f(u) ∈ Z, of coefficientε, applied tof(v)∈Z as
(f∗δ)fε(u)f(v) = f(δεuv) It is easy to check that this is indeed a dilation structure.
In particular we may consider to transport a dilation structure by one of its dilations. Visually, this corresponds to transporting the atlas representing a dilation structure on X to a neighbourhood of one of its points. It is like a scale reduction of the whole territory (X, d) to a smaller set.
Inversely, we may transport the (restriction of the) dilation structure (X, d, δ) from Vε(x) to U(x), by using δxε−1 as the transport function f. This is like a magnification of the ”infinitesimal neighbourhood”Vε(x). (This neighbourhood
is infinitesimal in the sense that we may consider ε as a variable, going to 0 when needed. Thus, instead of one neighbourhoodVε(x), there is a sequence of
them, smaller and smaller).
This is useful, because it allows us to make ”infinitesimal statements”, i.e. statements concerning this sequence of magnifications, asε→0.
Let us compute then the magnified dilation structure. We should also rescale the distance on Vε(x) by a factor 1/ε. Let us compute this magnified dilation
structure:
- the space isU(x)
- for anyu, v∈U(x) the (transported) distance between them is dxε(u, v) = 1 εd(δ x εu, δ x εv)
- for any u, v ∈U(x) and scale parameterµ∈(0,1) (we could take µ >0 but then we have to be careful with the domains and codomains of these new dilations), the transported dilation based atu, of coefficientµ, applied tov, is
δxε−1δ
δxεu
ε δεxv (8.1.3)
It is visible that working with such combinations of dilations becomes quickly difficult. This is one of the reasons of looking for more graphical notations. Important remark. This sequence of magnified dilation structures, around x, at scale ε, with the parameterε seen as a variable converging to 0, will be later named ”chora”, or placexat scale varepsilon.
In particular, the transported dilation (8.1.3) will later appear as an ”ele- mentary chora”.
Definition 8.2 Let (X, d, δ)be a dilation structure. A property
P(x1, x2, x3, ...)
is true for x1, x2, x3, ...∈X sufficiently close if for any compact, non empty set
K⊂X, there is a positive constantC(K)>0such thatP(x1, x2, x3, ...)is true for any x1, x2, x3, ...∈K with d(xi, xj)≤C(K).
For a dilation structure the metric tangent spaces have the algebraic struc- ture of a normed group with dilations.
We shall work further with local groups, which are spaces endowed with a locally defined operation which satisfies the conditions of a uniform group. See section 3.3 [2] for details about the definition of local groups.