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Journal of Generalized Lie Theory and Applications
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Dilatation structures I. Fundamentals

Marius BULIGA*

”Simion Stoilow” Institute of Mathematics of the Romanian Academy, P.O. BOX 1-764, RO 014700 Bucure¸sti, Romania

*Corresponding Author:
Marius BULIGA
”Simion Stoilow” Institute of Mathematics of the Romanian Academy
P.O. BOX 1-764, RO 014700 Bucure
sti, Romania
E-mail: [email protected]

Visit for more related articles at Journal of Generalized Lie Theory and Applications

Abstract

A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is a part of a series of papers which show that such a structure allows to do non-commutative analysis, in the sense of differential calculus, on a large class of metric spaces, some of them fractals. We also describe a formal, universal calculus with binary decorated planar trees, which underlies any dilatation structure.

1 Introduction

The purpose of this paper is to introduce dilatation structures on metric spaces. A dilatation structure is a concept in between a group and a differential structure. Any metric space (X, d) endowed with a dilatation structure has an associated tangent bundle. The tangent space at a point is a conical group, that is the tangent space has a group structure together with a one-parameter group of automorphisms. Conical groups generalize Carnot groups, i.e nilpotent groups endowed with a graduation. Each dilatation structure leads to a non-commutative differential calculus on the metric space (X, d).

There are several important papers dedicated to the study of extra structures on a metric space which allows to do a reasonable analysis in such spaces, like Cheeger [6] or Margulis- Mostow [10, 11].

The constructions proposed in this paper first appeared in connection to problems in analysis on sub-riemannian manifolds. Parts of this article can be seen as a rigorous formulation of the considerations in the last section of Bella¨ıche [1].

A dilatation structure is simply a bundle of semigroups of (quasi-)contractions on the metric space (X, d), satisfying a number of axioms. The tangent bundle structure associated with a given dilatation structure on the metric space (X, d) is obtained by a passage to the limit procedure, starting from an algebraic structure which lives on the metric space.

With the help of the dilatation structure we construct a bundle (over the metric space) of (local) operations: to each imageand parameter ", for simplicity hereimage, there is a natural non-associative operation

image

where U(x) is a neighbourhood of x. The non-associativity of this operation is controlled by the parameter ". As " goes to 0 the operation imageconverges to a group operation on the tangent space of (X, d) at x.

Denote by image the dilatation based at image, of parameter ". The bundle of operations satisfies a kind of weak associativity, even if for any fixed image the operation image is non-associative. The weak associativity property, named also shifted associativity, is

image

for any image and any image sufficiently close to X. We shall describe also other objects (like a function satisfying a shifted inverse property) and algebraic identities related to the dilatation structure and the induced bundle of operations.

We briefly describe further the contents of the paper. In section 2 we give motivational examples of dilatation structures. Basic notions and results of metric geometry and groups endowed with dilatations are mentioned in section 3.

In section 4 we introduce a formalism based on decorated planar binary trees. This formalism will be used to prove the main results of the paper. We show that, from an algebraic point of view, dilatation structures (more precisely the formalism in section 4) induce a bundle of one parameter deformations of binary operations, which are not associative, but shifted associative. This is a structure which bears resemblance with the tangent bundle of a Lie group, but it is more general.

Section 5, 6 and 7 are devoted to dilatation structures. These sections contain the main results of the paper. After we introduce and explain the axioms of dilatation structures, we describe several key metric properties of such a structure, in section 5. In section 6 we prove that a dilatation structure induces a valid notion of tangent bundle. In section 7 we explain how a dilatation structure leads to a differential calculus.

Section 8 is made of two parts. In the first part we show that dilatation structures induce differential structures, in a generalized sense. In the second part we turn to conical groups and we prove the curious result that, even if in a conical group left translations are smooth but right translations are generically non differentiable, the group operation is smooth if we well choose a dilatation structure.

2 Motivation

We start with a trivial example of a dilatation structure, then we briefly explain the occurence of such a structure in more unusual situations.

There is a lot of structure hiding in the dilatations of imageFor this space, the dilatation based at x, of coefficient image, is the function

image

For fixed x the dilatations based at x form a one parameter group which contracts any bounded neighbourhood of x to a point, uniformly with respect to x.

Dilatations behave well with respect to the euclidean distance d, in the following sense: for any image and any image we have

image

This shows that from the metric point of view the space image is a metric cone, that is image looks the same at all scales.

Moreover, let image be a function andimage .The function f is differentiable in x if there is a linear transformation A (that is a group morphism which commutes with dilatations based at the neutral element 0) such that the limit.

image           (21)

is uniform with respect to v in bounded neighbourhood of x. Really, let us calculate

image

This shows that we get the usual definition of differentiability.

The relation (2.1) can be put in another form, using the euclidean distance:

image

uniformly with respect to v in bounded neighbourhood of x. Here

image

In conclusion, dilatations are the fundamental object for doing differential calculus on image. Even the algebraic structure of image is encoded in dilatations. Really, we can recover the operation of addition from dilatations. It goes like this: for image and image define

image

For fixed imagethe functions image are inverse one to another, but we don’t insist on this for the moment (see Proposition 3).

What is the meaning of these functions? Let us calculate

image

In the same way we get

image

As image we have the following limits:

image

uniform with respect to imagein bounded sets. The function image is a group operation, namely the addition operation translated such that the neutral element is x. Thus, for x = 0, we recover the group operation. The function image is the inverse function, and imageis the difference function.

Notice that for fixed image the function imageis not a group operation, first of all because it is not associative. Nevertheless, this function satisfies a shifted associativity property, namely (see Proposition 5).

image

Also, the inverse functionimage is not involutive, but shifted involutive (Proposition 4),

image

These and other properties of dilatations allow to recover the structure of the tangent bundle of image,which is trivial in this case.

Let us go to more elaborate examples. We may look to a riemannian manifold M, which is locally a deformation of image. We can use charts for transporting (locally) the dilatation structure from image to the manifold. All the previously described metric and algebraic properties will hold in this situation, in a weaker form. For example the riemannian distance is no longer scalling invariant, but we still have

image

uniform limit with respect tu x, u, v in (small) bounded sets. Here dx is an euclidean distance which can be identified with the distance in the tangent space of M at x, induced by the metric tensor at x. In the same way we can construct the algebraic structure of the tangent space at x, using the functions image We will have a differentiability notion coming from the dilatations transported by the chart.

If we change charts or the riemannian metric then the dilatation structure will change too, but not very much, essentially because the change of charts is smooth, therefore we are still able to say what are tangent spaces and to describe their algebraic structure.

Let us go further with more complex examples. Consider the Heisenberg group H(n). As a set image We shall use the following notation: an element of H(n) will be denoted by image withimage The group operation is

image

where imageis the canonic symplectic 2-form on image

The group H(n) is nilpotent, in fact a 2 graded Carnot group. This means that H(n) is nilpotent and that it admits a one-parameter group of isomorphisms

image

These are dilatations, more precisely we can construct dilatations based at ˜x by the formula

image

We may also put a scalling invariant distance on H(n), for example as follows:

image

We can repeat step by step the constructions explained before in this situation. There are some differences though.

First of all, from the metric point of view, (H(n), d) is a fractal space, in the sense that the Hausdorff dimension of this space is equal to 2n+2, therefore strictly greater than the topological dimension, which is 2n+1. Second, the differential of a function defined by the dilatations is not the usual differential, but an essentially different one, called Pansu derivative (see [13]). This is part of a very active area of research in geometric analysis (among fundamental references one may cite [13, 6, 10, 11, 7]). A spectacular application of Pansu derivative was to prove a Rademacher theorem which in turn implies deep results about Mostow rigidity. The theory applies to general Carnot groups.

The Heisenberg group is not commutative. It is in fact the model for the tangent space of a contact metric manifold, as the euclidean image is the model of the tangent space of a riemannian manifold. We enter here in the realm of sub-riemannian geometry (see for example [1,9]). In a future paper we shall deal with dilatation structures for sub-riemannian manifolds. An important problem in sub-riemannian geometry is to have good tangent bundle structures, which in turn allow us to prove basic theorems, like Poincar´e inequality, Rademacher or Stepanov theorems.

We may even go further and find dilatation structures related with rectifiable sets, or with some self-similar sets. This is not the purpose of this paper though. In the sequel we shall define and study fundamental properties of dilatation structures.

3 Basic notions

We denote by image a relation and we writeimage Therefore we may haveimage

The domain of f is the set of image such that there isimage withimage. We denote the domain by image The image of f is the set of imagesuch that there is image withimage We denote the image by im f.

By convention, when we state that a relation image is true, it means thatimage is true for any choice ofimage such thatimage

In a metric space image the ball centered at image and radius r > 0 is denoted by B(x, r). If we need to emphasize the dependence on the distance d then we shall use the notation Bd(x, r).

In the same way,imageand image denote the closed ball centered at x, with radius r.

We shall use the following convenient notation: by image we mean a positive function such that image.

3.1 Gromov-Hausdorff distance

There are several definitions of distances between metric spaces. For this subject see [5] (Section 7.4), [8] (Chapter 3) and [7].

We explain now a well-known alternative definition of the Gromov-Hausdorff distance, up to a multiplicative factor

Definition 1. Let image be a pair of locally compact pointed metric spaces andimage We shall say that μ is admissible if there is a relation image such that

1. dom ρ is μ-dense in X1,

2. im ρ is μ-dense in X2,

3. image

4. for all image we have

image    (3.1)

The Gromov-Hausdorff distance between image is the infimum of admissible numbers μ.

Denote by image the isometry class ofimage that is the class of spacesimage such that it exists an isometryimage with the propertyimage Note that ifimage is isometric with image then they have the same diameter.

The Gromov-Hausdorff distance is in fact almost a distance between isometry classes of pointed metric spaces. Indeed, if two pointed metric spaces are isometric then the Gromov- Hausdorff distance equals 0. The converse is also true in the class of compact (pointed) metric spaces [8] (Proposition 3.6).

Moreover, if two of the isometry classes image have (representants with) diameter at most equal to 3, then the triangle inequality is true. We shall use this distance and the induced convergence for isometry classes of the form image

3.2 Metric profiles. Metric tangent space

We shall denote by CMS the set of isometry classes of pointed compact metric spaces. The distance on this set is the Gromov distance between (isometry classes of) pointed metric spaces and the topology is induced by this distance.

To any locally compact metric space we can associate a metric profile [3, 4].

Definition 2. The metric profile associated to the locally metric space (M, d) is the assignment (for small enough image image

We can define a notion of metric profile regardless to any distance.

Definition 3. A metric profile is a curve imagesuch that

(a) it is continuous at 0,

(b) for any imageand image we have

image

The function image may change with b. We used the notations

image

The metric profile is nice if

image

Imagine that 1/b represents the magnification on the scale of a microscope. We use the microscope to study a specimen. For each image the information that we get is the table of distances of the pointed metric space image

How can we know, just from the information given by the microscope, that the string of ”images” that we have corresponds to a real specimen? The answer is that a reasonable check is the relation from point (b) of the definition of metric profiles 3.

Really, this point says that starting from any magnification 1/b, if we further select the ball image in the snapshotimage then the metric spaceimage looks approximately the same as the snapshot image That is: further magnification by ε of the snapshot (taken with magnification) b is roughly the same as the snapshot image. This is of course true in a neighbourhood of the base point xb.

The point (a) from the Definition 3has no other justification than Proposition 1 in next subsection.

We rewrite definition 1 with more details, in order to clearly understand what is a metric profile. For any image and for anyimage such that for anyimageimage there exists a relationimage such that

image image  (3.2)

In the microscope interpretation, if image

means that x and u represent the same ”real” point in the specimen.

Therefore a metric profile gives two types of information:

• a distance estimate like (3.2) from point 4,

• an ”approximate shape” estimate, like in the points 1–3, where we see that two sets, namely the balls image are approximately isometric.

The simplest metric profile is one with image In this case we see that imageis approximately an " dilatation with base point x.

This observation leads us to a particular class of (pointed) metric spaces, namely the metric cones.

Definition 4. A metric cone (X, d, x) is a locally compact metric space (X, d), with a marked point imagesuch that for any image we have

image

Metric cones have dilatations. By this we mean the following

Definition 5. Let (X, d, x) be a metric cone. For anyimage a dilatation is a functionimage such that

image

The existence of dilatations for metric cones comes from the definition 4. Indeed, dilatations are just isometries from image

Metric cones are good candidates for being tangent spaces in the metric sense.

Definition 6. A (locally compact) metric space (M, d) admits a (metric) tangent space in imageif the associated metric profile image (as in definition 2) admits a prolongation by continuity in " = 0, i.e if the following limit exists:

image       (3.3)

The connection between metric cones, tangent spaces and metric profiles in the abstract sense is made by the following proposition.

Proposition 1. The associated metric profile image of a metric space (M, d) for a fixed image is a metric profile in the sense of the definition 3 if and only if the space (M, d) admits a tangent space in x. In such a case the tangent space is a metric cone.

Proof. A tangent space [V, dv, v] exists if and only if we have the limit from the relation (3.3). In this case there exists a prolongation by continuity toimageof the metric profile image The prolongation is a metric profile in the sense of definition 3. Indeed, we have still to check the property (b). But this is trivial, because for any image sufficiently small, we have

image

where image

Finally, let us prove that the tangent space is a metric cone. For any imagewe have

image

Therefore

image

3.3 Groups with dilatations. Virtual tangent space

In section 6 we shall see that metric tangent spaces sometimes have a group structure which is compatible with dilatations. This structure, of a group with dilatations, is interesting by itself. The notion has been introduced in [2]; we describe it further.

We start with the following setting: G is a topological group endowed with an uniformity such that the operation is uniformly continuous. The description that follows is slightly non canonical, but is nevertheless motivated by the case of a Lie group endowed with a Carnot-Caratheodory distance induced by a left invariant distribution.

We introduce first the double of G, as the group G(2) = G × G with operation

image

The operation on the group G, seen as the function

image

is a group morphism. Also the inclusions:

image

are group morphisms.

Definition 7. 1. G is a uniform group if we have two uniformity structures, on G and G2 such that op,image are uniformly continuous.

2. A local action of a uniform group G on a uniform pointed space (X, x0) is a function

image such that

(a) the map image is uniformly continuous from G×X (with product uniformity) to X,

(b) for any image there isimage such that for anyimage make sense andimage

3. Finally, a local group is an uniform space G with an operation defined in a neighbourhood ofimage which satisfies the uniform group axioms locally.

Note that a local group acts locally at left (and also by conjugation) on itself.

This definition deserves an explanation. An uniform group, according to the Definition 7, is a group G such that left translations are uniformly continuous functions and the left action of G on itself is uniformly continuous too. In order to precisely formulate this we need two uniformities: one on G and another on G × G.

These uniformities should be compatible, which is achieved by saying that imageare uniformly continuous. The uniformity of the group operation is achieved by saying that the op morphism is uniformly continuous.

Definition 8. A group with dilatations (G) is a local uniform group G with a local action of image (denoted by δ), on G such that

H0. the image exists and is uniform with respect to x in a compact neighbourhood of the identity e.

H1. the limit

image

is well defined in a compact neighbourhood of e and the limit is uniform.

H2. the following relation holds:

image

where the limit from the left hand side exists in a neighbourhood of e and is uniform with respect to x.

These axioms are the prototype of a dilatation structure.

The ”infinitesimal version” of an uniform group is a conical local uniform group.

Definition 9. A conical group N is a local group with a local action of (0,+∞) by morphisms image such that image for any x in a neighbourhood of the neutral element e.

Here comes a proposition which explains why a conical group is the infinitesimal version of a group with dilatations.

Proposition 2. Under the hypotheses H0, H1, H2 (G, β, δ) is a conical group, with operation β and dilatations δ.

Proof. All the uniformity assumptions allow us to change at will the order of taking limits. We shall not insist on this further and we shall concentrate on the algebraic aspects.

We have to prove the associativity, existence of neutral element, existence of inverse and the property of being conical.

For the associativity imagewe calculate

image

We take image and get

image

In the same way

image

and again taking image we obtain

image

The neutral element is e, from H0 (first part) it follows thatimage The inverse of x is x−1, by a similar argument:

image

and taking image we obtain

image

Finally, βhas the property

image

which comes from the definition of β and commutativity of multiplication in (0,+∞). This proves that image is conical.

In a sense image is the tangent space of the group with dilatations imageat e. We can act with the conical groupimageon image.Indeed, let us denote by image the commutator of two transformations. For the group G we shall denote by image the left translation and by image. The preceding proposition tells us that image acts locally by left translations on G. We shall call the left translations with respect to the group operation ”infinitesimal”. These infinitesimal translations admit an interesting commutator representation

image        (3.4)

Definition 10. The group image formed by all transformationsimage is called the virtual tangent space at e to G.

As local groups, image and image are isomorphic. We can easily define dilatations on image, by conjugation with dilatations imageReally, we see that

image

The virtual tangent space image at imageis obtained by translating the group operation and the dilatations from e to x. This means: define a new operation on G by

image

The group G with this operation is isomorphic to G with old operation and the left translation image is the isomorphism. The neutral element is x. Introduce also the dilatations based at x by

image

Then image with the group of dilatationsimage satisfy the Axioms H0, H1, H2. Define then the virtual tangent space image

4 Binary decorated trees and dilatations

We want to explore what happens when we make compositions of dilatations (which depends also on image ). The " variable apart, any dilatationimage is a function of two arguments: x and y, invertible with respect to the second argument. The functions we can obtain when composing dilatations are difficult to write, that is why we shall use a tree notation.

4.1 The formalism

Let X be a non empty set and T (X) be a class of binary planar trees with leaves in X and all nodes decorated with two colors {○, •}. The empty tree, that is the tree with no nodes or leaves, belongs to T (X). For any x εX we accept that there is a tree in T (X) with no nodes and with x as the only leaf. That is image.

For any color image let ¯a be the opposite color. The colors ○ and • are codes for the symbols ε and ε−1

The relation imageis an equivalence relation on T (X), taken as a primitive notion for the axioms which will follow..

The equivalence class of of a tree image is denoted byimage In various diagrams that will follow we shall use the notation image for saying that T is the equivalence class of P. For any

image

Axiom T0. For any image the trees

image

The equivalence class of image is denoted byimage that is we have

image

Axiom T1. Consider any trees image and any colors a, b such that the trees from the right hand sides of relations below belong to T (X). Then the trees from the left hand sides of relations below belong to T (X) and we have

image

Here, in all diagrams, the symbol image means that the node colored with a is grafted at an arbitrary leaf of the tree S.

The second axiom expresses the fact that the dilatation (of any coefficient image has x as fixed point, that is image

Axiom T2. For any image the tree image belongs to T (X). Moreover, consider any tree imageand anyimage. Then the trees from the left hand sides of relations below belong to T (X) and we have

image

that is the equivalence class of x is the same as the equivalence class of image and the equivalence class of image As in Axiom T1, the symbolimage means that the root of the tree P is grafted at an arbitrary leaf of the tree S.

Definition 11. We define the difference, sum and inverse trees as follows:

(a) the difference tree image is given by the relation

image

(b) the sum tree image is given by the relation

image

(c) the inverse tree image is given by the relation

image

The next axiom states that T0, T1, T2 are sufficient for determining the class T (X) and the equivalence relation image.

Axiom T3. The class T (X) is the smallest class of trees obtained by grafting of trees listed in Axiom T0, and satisfying Axioms T1, T2. Moreover, two trees from T (X) are equivalent if and only if they can be proved equivalent after a finite string of applications of Axioms T1, T2.

4.2 First consequences

We shall use the axioms in order to obtain results that we shall use later, for dilatation structures.

Proposition 3. For any x, u, y and v we have

image

Proof. We prove (a) by computations using the definition 11 of the sum and difference trees, and Axiom T1 several times.

image

For (b) we proceed in the same way:

image

Proposition 4. We have the relations

image        (4.4)

image    (4.5)

Proof. Graphically, the relation (4.3) is

image

This is true by Axiom T1.

The relation (4.4) is

image

This is true by Axiom T2.

We prove the relation (4.5) by a string of equalities, starting from the left hand side to the right:

image

Here we have used the Axiom T1 several times.

The relation (4.5) in last proposition shows that the ”inverse function”image " is not involutive, but shifted involutive.

The next proposition shows that the function image satisfies a shifted associativity property.

Proposition 5. We have the following relations:

image      (4.6)

image       (4.7)

Proof. Graphically, the relation (4.6) is

image

This is true by Axiom T1.

The relation (4.7) is is equivalent to (4.6), by Proposition 3. We can also give a direct proof by graphically representing the relation

image

This is true by the Axiom T1.

5 Dilatation structures

The space (X, d) is a complete, locally compact metric space. This means that as a metric space (X, d) is complete and that small balls are compact.

5.1 Axioms of dilatation structures

The axioms of a dilatation structure (X, d, δ) are listed further. The first axiom is merely a preparation for the next axioms. That is why we counted it as Axiom 0.

A0. Depending on the parameter image, dilatations are objects having the following description.

For any image the dilatations are functions

image

All such dilatations are homeomorphisms (invertible, continuous, with continuous inverse). We suppose that there is 1 < A such that for any imagewe have

image

We suppose that for all image, we have

image

For image the associated dilatation

image

is an injective, continuous, with continuous inverse on the image. We shall suppose that image is open,

image

and that for all imageandimage we have

image

We remark that we have the following string of inclusions, for any image image

A further technical condition on the sets image andimage will be given just before the Axiom A4. (This condition will be counted as part of Axiom A0.)

A1. We have image for any point x. We also haveimage for any image.

Let us define the topological space

image

with the topology inherited from the product topology on image Consider also image the closure of image with product topology. The function

image

defined by image is continuous. Moreover, it can be continuously extended toimage and we have

image

A2. For any image image

A3. For any x there is a function image defined for any u, v in the closed ball (in distance d)image such that

image

uniformly with respect to x in compact set.

Remark 12. The ”distance” dx can be degenerated. That means: there might be v,w,εimage such that dx(v,w) = 0 but v 6= w. We shall use further the name ”distance” for dx, essentially by commodity, but keep in mind the possible degeneracy of dx.

For the following axiom to make sense we impose a technical condition on the co-domains image for any compact setimage there areimage andimage such that for all image and allimage,we have

image

With this assumption the following notation makes sense:

image        (5.1)

The next axiom can now be stated:

A4. We have the limit

image

uniformly with respect to x, u, v in compact set.

Note that with the tree notation we may identify (5.1) with the difference tree from Definition 11 (a).

Definition 13. A triple (X, d,δ) which satisfies A0, A1, A2, A3, but dx is degenerate for some image

,is called degenerate dilatation structure.

If the triple (X, d,δ) satisfies A0, A1, A2, A3 and dx is non-degenerate for any image, then we call it a weak dilatation structure.

If a weak dilatation structure satisfies A4 then we call it dilatation structure.

Note that it could be assumed, without great modification of the axioms, that

(a) we may replace (0,∞) by image , a topological separated commutative group endowed with a continuous group morphismimage with image Here (0,+∞) is taken as a group with multiplication. The neutral element of �� is denoted by 1. We use the multiplicative notation for the operation in image.

The morphism v defines an invariant topological filter on image(equivalently, an end). Really, this is the filter generated by the open sets image From now on we shall name this topological filter (end) by ”0” and we shall write image

The set image is a semigroup. We noteimage On the setimage we extend the operation on image by adding the rules 00 = 0 and "0 = 0 imagefor any image This is in agreement with the invariance of the end 0 with respect to translations in image.

In the Axioms A0, A1 we therefore may replaceimage , and so forth. (b) we may leave some flexibility in Axiom A1 for the choice of base point of the dilatation, in the sense that

(b) we may leave some flexibility in Axiom A1 for the choice of base point of the dilatation, in the sense that

image

uniformly with respect toimage compact set,

(c) we may relax the semigroup condition in the Axiom A2, in the sense: for any compact set image for anyimage image

(d) in the Axioms A3 and A4 we may replace image

We shall write the proofs of further results such that these work even if we modify the axioms in the sense explained above. We shall nevertheless use " and not ("), in order to avoid a too heavy notation.

The axioms, as given in this section, are said to be in strong form. With the modifications explained at points (a), (b), (c), (d) above, the axioms are said to be in weak form.

Further, axioms are taken in weak form with the notational conventions explained above, unless it is explicitely stated that some axiom has to be taken in strong form.

5.2 Dilatation structures, tangent cones and metric profiles

We shall explain now what the axioms mean. The first Axiom A1 is stating that the distance between image is negligible with respect toimage then this axiom is trivially satisfied. The second Axiom A2. states that in an approximate sense the transformations image form an action of image on X. As previously, if we suppose that

image

then this axiom is trivially satisfied.

Remark now that the binary tree formalism described in section 4 underlies and simplifies the calculus with dilatation structures. More precisely, we shall use the results in section 4 in the proof of theorems in the next section.

The notation with binary trees for composition of dilatations is not directly adapted for taking limits as imageAn extension of the formalism can be made in this direction, but this would add length to this paper, which is devoted to first properties of dilatation structures. We reserve the full description of the formalism for a future paper.

In Axiom A3 we take limits. In this subsection we shall look at dilatation structures from the metric point of view, by using Gromov-Hausdorff distance and metric profiles.

We state the interpretation of the Axiom A3 as a theorem. But before a definition: we denote by image the distance on

image

given by

image

Theorem 6. Let (X, d,δ) be a dilatation structure. The following are consequences of the Axioms A0 - A3 only:

(a) for all image such thatimage image

We shall say that dx has the cone property with respect to dilatations.

(b) The curve image is a metric profile.

Proof. (a) For imagewe have

image

Use now the Axioms A2 and A3 and pass to the limit with imageThis gives the desired equality.

(b)We have to prove that image is a metric profile. For this we have to compare two pointed metric spaces:

image

Let image such that

image

This means that

image

Further use the Axioms A1, A2 and the cone property proved before:

image

therefore,

image

It follows that for any image we can chooseimage such that

image

We want to prove that

image

This goes as follows:

image

In order to obtain the last estimate we used twice the Axiom A3. We proceed as follows:

image

This shows that the property (b) of a metric profile is satisfied. The property (a) is proved in the Theorem 7.

The following theorem is related to Mitchell [12] Theorem 1, concerning sub-riemannian geometry.

Theorem 7. In the hypothesis of theorem 6, we have the following limit:

image

Therefore if dx is a true (i.e. nondegenerate) distance, then (X, d) admits a metric tangent space in x.

Moreover, the metric profile image is almost nice, in the following sense. Let image Then we have the inclusion .

image

Moreover, the following Gromov-Hausdorff distance is of order image fixed (that is the modulus of convergence image does not depend on μ):

image

For another Gromov-Hausdorff distance we have the estimate

image whenimage

Proof. We start from the Axioms A0, A3 and we use the cone property. By A0, for image andimage there existimage such that

image

By the cone property we have

image

By A2 we have

image

This proves the first part of the theorem.

For the second part of the theorem take any image Then we have

image

Then there exists image such that for anyimage and u in the mentioned ball we have image

In this case we can take directly image and simplify the string of inequalities from the proof of Theorem 6, point (b), to get eventually the three points from the second part of the theorem.

6 Tangent bundle of a dilatation structure

In this section we shall use the calculus with binary decorated trees introduced in section 4, for a space endowed with a dilatation structure.

6.1 Main results

Theorem 8. Let (X, d, δ) be a dilatation structure. Then the ”infinitesimal translations”

image are dx isometries.

Proof. The first part of the conclusion of Theorem 7 can be written as follows:

image          (6.1)

as image

For image sufficiently small the points image are close one to another. Precisely, we have

image

Therefore, if we choose u, v,w such that image andimage then there isimage such that for all image we have

image

We apply the estimate (6.1) for the basepoint image to get

image

when image This can be written, using the cone property of the distance image like

image       (6.2)

as image By the Axioms A1, A3, the function

image

is an uniform limit of continuous functions, therefore uniformly continuous on compact sets. We can pass to the limit in the left hand side of the estimate (6.2), using this uniform continuity and Axioms A3, A4, to get the result.

Let us define, in agreement with definition 11 (b)

image

Corollary 9. If for any x the distance dx is non degenerate then there exists C > 0 such that for any x and u with image there exists a dx isometryimage image

uniformly with respect to x, u, v in compact set.

Proof. From Theorem 8 we know that image is a dx isometry. If dx is non degenerate then image is invertible. Let image be the inverse.

From Proposition 3 we know that image is the inverse of image.Therefore

image

From the uniformity of convergence in Theorem 8 and the uniformity assumptions in axioms of dilatation structures, the conclusion follows.

The next theorem is the generalization of Proposition 2. It is the main result of this paper.

Theorem 10. Let (X, d, δ) be a dilatation structure which satisfies the strong form of the Axiom A2. Then for any image is a conical group. Moreover, left translations of this group are dx isometries.

Proof. We start by proving that image is a local uniform group. The uniformities are induced by the distance d.

We shall use the general relations written in terms of binary decorated trees. According to relation (4.4) in Proposition 4, we can pass to the limit with image and define

image

From relation (4.5) we get (after passing to the limit with image image

We shall see that imageis the inverse of u. Relation (4.3) gives

image                  (6.3)

therefore relations (a), (b) from Proposition 3 give

image

Relation (4.7) from Proposition 5 gives

image

which shows that image is an associative operation. From (6.5), (6.4) we obtain that for any u, v

image

Remark that for any image indeed, this means that

image

Therefore x is a neutral element at left for the operation image. From the definition of invx, relation (6.3) and the fact that invx is equal to its inverse, we get that x is an inverse at right too: for any x, v we have

image

Replace now v by x in relations (6.7), (6.8) and prove that indeedimage is the inverse of u.

We still have to prove that imageas dilatations.In this reasoning we need the Axiom A2 in strong form.

Namely we have to prove that for any image we have

image

For this is sufficient to notice that

image

and pass to the limit as imageNotice that here we used the fact that dilatations image exactly commute (Axiom A2 in strong form).

Finally, left translations image isometries. Really, this is a straightforward consequence of Theorem 8 and corollary 9.

The conical group image can be regarded as the tangent space ofimage and denoted further by image

6.2 Algebraic interpretation

In order to better understand the algebraic structure of the sum, difference, inverse operations induced by a dilatation structure, we collect previous results regarding the properties of these operations, into one place.

Theorem 11. Let (X, d, δ) be a weak dilatation structure. Then, for any image we have

image

(d) The sum operation is shifted associative: for any u, v,w sufficiently close to x we have

image

(e) The difference, inverse and sum operations are related by

image

for any u, v sufficiently close to x

(f ) For any u, v sufficiently close to x and image we have

7 Dilatation structures and differentiability

7.1 Equivalent dilatation structures

Definition 14. Two dilatation structures image are equivalent if

(a) the identity image is bilipschitz and

(b) for any image there are functionsimage (defined forimage sufficiently close to x) such that

image

uniformly with respect to x, u in compact sets.

Proposition 12. Two dilatation structures image are equivalent if and only if

(a) the identity map id :image is bilipschitz and is bilipschitz and

(b) for any image there are functionsimage (defined forimage sufficiently close to x) such that

image

uniformly with respect to x, u in compact sets.

Proof. We make the notations

image

The relation (7.1) is equivalent to

image as image, uniformly with respect to x, u in compact sets. The conclusion follows after passing image

The next theorem shows a link between the tangent bundles of equivalent dilatation structures.

Theorem 13. Letimage be equivalent dilatation structures. Suppose that for any image the distance dx is non degenerate. Then for any image and any image sufficiently close to x we have:

image         (7.5)

The two tangent bundles are therefore isomorphic in a natural sense.

Proof. We notice first that the hypothesis is symmetric: if dx is non degenerate then image is non degenerate too. Really, this is straightforward from definition 14 (a) and Axiom A3 for the two dilatation structures.

For the proof of relation (7.5) is enough to remark that for " > 0 but sufficiently small we have image but sufficiently small we have

image

Really, with tree notation, let

image

The relation (7.6), written from right to left, is

image

But this is true by cancellations of dilatations and definitions of the operators image

7.2 Differentiable functions

Dilatation structures allow to define differentiable functions. The idea is to keep only one relation from definition 14, namely (7.1). We also renounce to uniform convergence with respect to x and u, and we replace this with uniform convergence in u and with a conical group morphism condition for the derivative.

First we need the natural definition below.

Definition 15. Let image be two conical groups. A functionimage is a conical group morphism if f is a group morphism and for any image we haveimage

The definition of derivative with respect to dilatations structures follows.

Definition 16. Let image be two dilatation structures andimage be a continuous function. The function f is differentiable in x if there exists a conical group morphism image defined on a neighbourhood of x with values in a neighbourhood of f(x) such that

image        (7.7)

The morphism Qx is called the derivative of f at x and will be sometimes denoted by Df(x).

The function f is uniformly differentiable if it is differentiable everywhere and the limit in (7.7) is uniform in x in compact sets.

This definition deserves a short discussion. Let image be two dilatation structures image and function differentiable in x. The derivative of f in x is a conical group morphism image which means that Df(x) is defined on a open set around x with values in a open set around f(x), having the following properties:

(a) for any u, v sufficiently close to x

image

(b) for any u sufficiently close to x and any image image

(c) the function Df(x) is continuous, as uniform limit of continuous functions. Indeed, the relation (7.7) is equivalent to the existence of the uniform limit (with respect to u in compact sets)

image

From (7.7) alone and axioms of dilatation structures we can prove properties (b) and (c). We can reformulate therefore the definition of the derivative by asking that Df(x) exists as an uniform limit (as in point (c) above) and that Df(x) has the property (a) above. From these considerations the chain rule for derivatives is straightforward.

A trivial way to obtain a differentiable function (everywhere) is to modify the dilatation structure on the target space.

Definition 17. Let (X, δ, d) be a dilatation structure and image be a bilipschitz and surjective function. We define then the transport of (X, δ, d) by f, named imageby

image

The relation of differentiability with equivalent dilatation structures is given by the following simple

Proposition 14. Letimagebe two dilatation structures and image be a bilipschitz and surjective function. The dilatation structures imageare equivalent if and only if f and f−1 are uniformly differentiable.

Proof. Straightforward from definitions 14 and 17.

8 Differential structure, conical groups and dilatation structures

In this section we collect some facts which relate differential structures with dilatation structures. We resume then the paper with a justification of the unusual way of defining uniform groups (definition 7) by the fact that the op function (the group operation) is differentiable with respect to dilatation structures which are natural for a group with dilatations.

8.1 Differential structures and dilatation structures

A differential structure on a manifold is an equivalence class of compatible atlases. We show here that an atlas induces an equivalence class of dilatation structures and that two compatible atlases induce the same equivalence class of dilatation structures.

Let image n-dimensional real manifold and A an atlas of this manifold. For each chart image we shall define a dilatation structure on W.

Suppose that image is convex (if not then take an open subset of W with this property). For image define the dilatation

image

Otherwise said, the dilatations in W are transported from image Equally, we transport on W the euclidean distance of image. We obviously get a dilatation structure on W.

If we have two charts image elonging to the same atlas A, then we have two equivalent dilatation structures on image Indeed, the atlas A is C1 therefore the distances (induced from the charts) are (locally) in bilipschitz equivalence. Denote byimage the dilatation obtained from the chart image. A short computation shows that (we use here the transition map image image

therefore, image, we have

image

A similar computation shows that Px also exists. The uniform convergence requirements come from the fact that we use a C1 atlas.

A similar reasoning shows that in fact two compatible atlases induce the same equivalence class of dilatation structures.

8.2 Conical groups and dilatation structures

In a group with dilatations (G, δ) we define dilatations based in any point image by

image             (8.1)

Definition 18. A normed group with dilatations image is a group with dilatations (G, δ) endowed with a continuous norm function image which satisfies (locally, in a neighbourhood of the neutral element e) the following properties:

image

It is easy to see that if image is a normed group with dilatations thenimage is a normed conical group. The norm image satisfies the stronger form of property (d) of Definition 18: for any image

Normed groups with dilatations can be encountered in sub-Riemannian geometry. Normed conical groups generalize the notion of Carnot groups.

In a normed group with dilatations we have a natural left invariant distance given by

image        (8.2)

Theorem 15. Let image be a locally compact normed group with dilatations. Then (G, δ, d) is a dilatation structure, where δ are the dilatations defined by (8.1) and the distance d is induced by the norm as in (8.2).

Proof. The Axiom A0 is straightforward from definition 7, definition 8, Axiom H0, and because the dilatation structure is left invariant, in the sense that the transport by left translations in G, according to Definition 17, preserves the dilatations δ. We also trivially have Axioms A1 and A2 satisfied.

For the Axiom A3 remark that

image

Denote image image

We have then:

image

Define the function

image

From Definition 8 Axioms H1, H2, and from definition 18 (d), we obtain that Axiom A3 is satisfied.

For the Axiom A4 we have to calculate

imageas " ! 0. Therefore the Axiom A4 is satisfied.

We remarked in the proof of the previous theorem that the transport by left translations in G, according to Definition 17, preserves the dilatation structure on G. This implies, according to Proposition 14, that left translations are differentiable. On the contrary, a short computation and examples from sub-Riemannian geometry indicate that right translations are not differentiable.

Nevertheless, the operation op is differentiable, if we endow the group G(2) = G × G with a good dilatation structure. This will justify the non standard way to define local uniform groups in Definition 7.

Start from the fact that if G is a local uniform group then G(2)is a local uniform group too. If G is also normed, with dilatations, then we can easily define a similar structure on G(2). Really, the norm onG(2)can be taken as

image

and dilatations

image

We leave to the reader to check that G(2) endowed with this norm and these dilatations is indeed a normed group with dilatations.

Theorem 16. Let image be a locally compact normed group with dilatations and letimage be the associated normed group with dilatation. Then the operation (op function) is differentiable.

Proof. We start from the formula (easy to check in G(2))

image

Then we have

image

Let us define

image

Then we have

image

The right hand side of this equality converges then to 0 as " ! 0. More precisely, we have image More precisely, we have

image

In particular, we have image which shows that the operation β is the differential of the operation op calculated in the neutral element of G(2).

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