Gromov's dimension comparison problem on Carnot groups

We solve Gromov's dimension comparison problem on Carnot groups equipped with a Carnot–Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot–Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. To cite this article: Z.M. Balogh et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Unrectifiability and rigidity in stratified groups

In the class of stratified groups endowed with a left invariant Carnot-Carathéodory distance, we give an algebraic characterization of purely unrectifiable groups and we study rigidity properties. The main feature of our approach is the use of a suitable area formula with respect to the Carnot-Carathéodory distance.Received: 8 March 2004     

Geodesics in Heisenberg Groups

We obtain equations of geodesic lines in Heisenberg groups H²ⁿ⁺¹and prove that the ideal boundary of the Heisenberg group H²ⁿ⁺¹is a sphere S^2n-1with a natural CR-structure and corresponding Carnot-Carathéodory metric, i.e. it is a one-point compactification of the Heisenberg group H^2n-1of the next dimension in a row.     

Counter example to the Besicovitch covering property for some Carnot groups equipped with their Carnot-Carathéodory metric

We construct a counter example to the Besicovitch covering property for some class of Carnot groups equipped with their Carnot - Carathéodory metric. The construction uses some regularity assumptions for minimal geodesics and for the distance function to the origin. As an example we show that groups of type H, thus including all Heisenberg groups, satisfy these assumptions. We prove in particular that, in groups of type H, the distance function to the origin is Pansu-differentiable outside the center with continuous Pansu-differential there.Mathematics Subject Classification (2000): 53C17 (22E25)     

Convexity and semiconvexity along vector fields

Given a family of vector fields we introduce a notion of convexity and of semiconvexity of a function along the trajectories of the fields and give infinitesimal characterizations in terms of inequalities in viscosity sense for the matrix of second derivatives with respect to the fields. We also prove that such functions are Lipschitz continuous with respect to the Carnot?CCarathéodory distance associated to the family of fields and have a bounded gradient in the directions of the fields. This extends to Carnot?CCarathéodory metric spaces several results for the Heisenberg group and Carnot groups obtained by a number of authors.     

A nonholonomic Laplace operator

In this paper first the Laplace operator on a completely nonholonomic Riemannian manifold is defined in an invariant manner and its properties are considered. The method presented for studying it, as well as for the study of other hypoelliptic operators, involves the use of the geometry of nonholonomic manifolds. The nonholonomic metric (Carnot-Carathéodory metric), the Carathéodory measure, and hypoharmonic functions are defined. A theorem on the comparison of the spectra is proved and the connection is established between the bases of eigenfunctions of the ordinary and nonholonomic Laplacians. Conjectures are formulated on the principal term of the spectral asymptotic expansion of the nonholonomic Laplacian, on the structure of the wave fronts, and on the propagation of singularities.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 96–108, 1990.     

A Fefferman-Poincaré type inequality for Carnot-Carathéodory vector fields

In this note we prove a Fefferman-Poincaré type inequality in spaces with metric induced by Carnot-Carathéodory vector fields.

     

Distributional solutions of Burgers' equation and intrinsic regular graphs in Heisenberg groups

In the present paper we will characterize the continuous distributional solutions of Burgers' equation as those which induce intrinsic regular graphs in the first Heisenberg group H¹≡R³, endowed with a left-invariant metric d_∞ equivalent to its Carnot-Carathéodory metric. We will also extend the characterization to higher Heisenberg groups Hn≡R2n+1.     

Function spaces on fractals

We construct function spaces, analogs of Hölder-Zygmund, Besov and Sobolev spaces, on a class of post-critically finite self-similar fractals in general, and the Sierpinski gasket in particular, based on the Laplacian and effective resistance metric of Kigami. This theory is unrelated to the usual embeddings of these fractals in Euclidean space, and so our spaces are distinct from the function spaces of Jonsson and Wallin, although there are some coincidences for small orders of smoothness. We show that the Laplacian acts as one would expect an elliptic pseudodifferential operator of order d+1 on a space of dimension d to act, where d is determined by the growth rate of the measure of metric balls. We establish some Sobolev embedding theorems and some results on complex interpolation on these spaces.     

The coarea formula for real‐valued Lipschitz maps on stratified groups

We establish a coarea formula for real‐valued Lipschitz maps on stratified groups when the domain is endowed with a homogeneous distance and level sets are measured by the Q – 1 dimensional spherical Hausdorff measure. The number Q is the Hausdorff dimension of the group with respect to its Carnot–Carathéodory distance. We construct a Lipschitz function on the Heisenberg group which is not approximately differentiable on a set of positive measure, provided that the Euclidean notion of differentiability is adopted. The coarea formula for stratified groups also applies to this function, where the Euclidean one clearly fails. This phenomenon shows that the coarea formula holds for the natural class of Lipschitz functions which arises from the geometry of the group and that this class may be strictly larger than the usual one. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Partial regularity for degenerate subelliptic systems associated with Hörmander’s vector fields

The paper is devoted to partial Hölder estimates for weak solutions of a class of degenerate subelliptic systems constructed by Hörmander’s vector fields. Assumptions on the systems are that coefficient matrix satisfies VMO condition in the sense of Carnot-Carathéodory distance in variables x for fixed u, and lower order terms satisfy the natural growth condition and the controllable growth condition, respectively.     

Derivations and Dirichlet forms on fractals

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.     

Dimension of Slices Through Fractals with Initial Cubic Pattern

In this paper, the Hausdorff dimension of the intersection of self-similar fractals in Euclidean space R~n generated from an initial cube pattern with an(n-m)-dimensional hyperplane V in a fixed direction is discussed. The authors give a sufficient condition which ensures that the Hausdorff dimensions of the slices of the fractal sets generated by "multirules" take the value in Marstrand's theorem, i.e., the dimension of the self-similar sets minus one. For the self-similar fractals generated with initial cube pattern, this sufficient condition also ensures that the projection measure μVis absolutely continuous with respect to the Lebesgue measure L~m. When μV《 L~m, the connection of the local dimension ofμVand the box dimension of slices is given.     

Optimal mass transportation in the Heisenberg group

In this paper we consider the problem of optimal transportation of absolutely continuous masses in the Heisenberg group H_n, in the case when the cost function is either the square of the Carnot-Carathéodory distance or the square of the Korányi norm. In both cases we show existence and uniqueness of an optimal transport map. In the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function. In the latter case the proof requires some fine properties of BV functions in the Heisenberg group.     

Old wine in fractal bottles I: Orthogonal expansions on self-referential spaces via fractal transformations

Our results and examples show how transformations between self-similar sets may be continuous almost everywhere with respect to measures on the sets and may be used to carry well known notions from analysis and functional analysis, for example flows and spectral analysis, from familiar settings to new ones. The focus of this paper is on a number of surprising applications including what we call fractal Fourier analysis, in which the graphs of the basis functions are Cantor sets, discontinuous at a countable dense set of points, yet have good approximation properties. In a sequel, the focus will be on Lebesgue measure-preserving flows whose wave-fronts are fractals. The key idea is to use fractal transformations to provide unitary transformations between Hilbert spaces defined on attractors of iterated function systems.     

Local Metric Properties of Level Surfaces on Carnot–Carathéodory Spaces

Doklady Mathematics - An adequate local metric characteristic is introduced for level sets of $$C_{H}^{1}$$ -mappings of Carnot manifolds to Carnot–Carathéodory spaces. Moreover, for...     

Multivalued fractals in <Emphasis Type="Italic">b</Emphasis>-metric spaces

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems theory in several topics of applied sciences. It is known that examples of fractals and multivalued fractals are coming from fixed point theory for single-valued and multivalued operators, via the so-called fractal and multi-fractal operators. On the other hand, the most common setting for the study of fractals and multi-fractals is the case of operators on complete or compact metric spaces. The purpose of this paper is to extend the study of fractal operator theory for multivalued operators on complete b-metric spaces.     

A certain family of self-similar sets

In this paper we consider an one-parameter family of iterated function systems. For every value of the parameter we find the set of top addresses. We prove that this set is a countable disjoint union of self-similar sets and calculate its Hausdorff dimension.     

Iterated function systems with non-distinct fixed points

The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate the properties of the Hausdorff dimension of self-similar sets such that some functions in the corresponding iterated function system share the same fixed point. Then it is not possible to apply directly known techniques. In this paper we are going to calculate the Hausdorff dimension for almost all contracting parameters and calculate the proper dimensional Hausdorff measure of the attractor.     

Spectral distance on the circle

A building block of non-commutative geometry is the observation that most of the geometric information of a compact Riemannian spin manifold M is encoded within its Dirac operator D. Especially via Connes' distance formula one is able to extract from the spectral properties of D the geodesic distance on M. In this paper we investigate the distance d encoded within a covariant Dirac operator on a trivial U(n)-fiber bundle over the circle with arbitrary connection. It turns out that the connected components of d are tori whose dimension is given by the holonomy of the connection. For n=2 we explicitly compute d on all the connected components. For n?2 we restrict to a given fiber and find that the distance is given by the trace of the module of a matrix. The latest is defined by the holonomy and the coordinate of the points under consideration. This paper extends to arbitrary n and arbitrary connection the results obtained in a previous work for U(2)-bundle with constant connection. It confirms interesting properties of the spectral distance with respect to another distance naturally associated to connection, namely the horizontal or Carnot-Carathéodory distance dH. Especially in case the connection has irrational components, the connected components for d are the closure of the connected components of dH within the Euclidean topology on the torus.