Haar measure is not approximable by balls.
Haar measure on linear groups over local skew fields.
Haar null and non-dominating sets
We study the σ-ideal of Haar null sets on Polish groups. It is shown that on a non-locally compact Polish group with an invariant metric this σ-ideal is closely related, in a precise sense, to the σ-ideal of non-dominating subsets of . Among other consequences, this result implies that the family of closed Haar null sets on a Polish group with an invariant metric is Borel in the Effros Borel structure if, and only if, the group is locally compact. This answers a question of Kechris. We also obtain...
Hadamard-type inequalities for quasiconvex functions.
Hajłasz-Sobolev type spaces and -energy on the Sierpinski gasket.
Hake's property of a multidimensional generalized Riemann integral
Halbborelsche Funktionen und extreme Ableitungen
Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold
We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....
Hamiltonian loops from the ergodic point of view
Let be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold . A loop is called strictly ergodic if for some irrational number the associated skew product map defined by is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected...
Happy fractals and some aspects of analysis on metric spaces.
There has been a lot of interest and activity along the general lines of analysis on metric spaces recently, as in [2], [3], [26], [40], [41], [46], [48], [49], [51], [82], [83], [89], for instance. Of course this is closely related to and involves ideas concerning spaces of homogeneous type, as in [18], [19], [66], [67], [92], as well as sub-Riemannian spaces, e.g., [8], [9], [34], [47], [52], [53], [54], [55], [68], [70], [72], [73], [84], [86], [88]. In the present survey we try to give an introduction...
Harmonic analysis and the geometry of subsets of Rn.
This subject has several natural points of view, but we shall start with the one that corresponds to the following question: Is there something like Littlewood-Paley theory which is useful for analyzing the geometry of subsets of Rn, in much the same way that traditional Littlewood-Paley theory is good for analyzing functions and operators?
Harmonic analysis with respect to almost invariant measures
Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, II
Harmonic measure of some Cantor type sets.
Harmonic measures versus quasiconformal measures for hyperbolic groups
We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.
Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval II
We construct examples of expanding piecewise monotonic maps on the interval which have a closed topologically transitive invariant subset A with Darboux property, Hausdorff dimension d ∈ (0,1) and zero d-dimensional Hausdorff measure. This shows that the results about Hausdorff and conformal measures proved in the first part of this paper are in some sense best possible.
Hausdorff and conformal measures for expanding piecewise monotonic maps of the interval
Let A be a topologically transitive invariant subset of an expanding piecewise monotonic map on [0,1] with the Darboux property. We investigate existence and uniqueness of conformal measures on A and relate Hausdorff and conformal measures on A to each other.
Hausdorff and Fourier dimension
There is no constraint on the relation between the Fourier and Hausdorff dimension of a set beyond the condition that the Fourier dimension must not exceed the Hausdorff dimension.
Hausdorff and packing dimensions for ergodic invariant measures of two-dimensional Lorenz transformations
We extend the notions of Hausdorff and packing dimension introducing weights in their definition. These dimensions are computed for ergodic invariant probability measures of two-dimensional Lorenz transformations, which are transformations of the type occuring as first return maps to a certain cross section for the Lorenz differential equation. We give a formula of the dimensions of such measures in terms of entropy and Lyapunov exponents. This is done for two choices of the weights using the recurrence...
Hausdorff and packing measure for thick solenoids
For a linear solenoid with two different contraction coefficients and box dimension greater than 2, we give precise formulas for the Hausdorff and packing dimensions. We prove that the packing measure is infinite and give a condition necessary and sufficient for the Hausdorff measure to be positive, finite and equivalent to the SBR measure. We also give analogous results, generalizing [P], for affine IFS in ℝ².