This paper develops a general theory of uniform probability for compact metric spaces. Special cases of uniform probability include Lebesgue measure, the volume element on a Riemannian manifold, Haar measure, and various fractal measures (all suitably normalized). This paper first appeared fall of 1990 in the Journal of Theoretical Probability, vol. 3, no. 4, pp. 611—626. The key words by which...

We show that Haar measure is a unique measure on a torus or more generally a solenoid X invariant under a not virtually cyclic totally irreducible Zd-action by automorphisms of X such that at least one element of the action acts with positive entropy. We also give a corresponding theorem in the nonirreducible case. These results have applications regarding measurable factors and joinings of the...

We study the rigidity properties of a class of algebraic Z-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide examples of non-isomorphic Z-actions with all their Z-sub-actions isomorphic. The proofs use lexicographic half-spa...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2012-13/06c cpt ab gps.pdf] 1. Approximate identities on topological groups 2. Uniqueness of invariant measure 3. Simultaneous eigenfunctions for integral operators 4. Simultaneous eigenfunctions are characters The spectral theory for normal compact operators on Hil...

Let Γ be a dense subgroup of a simply connected nilpotent Lie group G generated by a finite symmetric set S. We consider the n-ball Sn for the word metric induced by S on Γ. We show that Sn (with uniform measure) becomes equidistributed on G with respect to the Haar measure as n tends to infinity. We also prove the analogous result for random walk averages.

In this paper, an efficient numerical scheme based on uniform Haar wavelets is used to solve the non-planar Burgers equation. The quasilinearization technique is used to conveniently handle the nonlinear terms in the non-planar Burgers equation. The basic idea of Haar wavelet collocation method is to convert the partial differential equation into a system of algebraic equations that involves a ...

Let Γ be a non-elementary subgroup of SL2(Z). If μ is a probability measure on T which is Γ-invariant, then μ is a convex combination of the Haar measure and an atomic probability measure supported by rational points. The same conclusion holds under the weaker assumption that μ is ν-stationary, i.e. μ = ν ∗ μ, where ν is a finitely supported probability measure on Γ whose support supp(ν) genera...

Hankel operators lie at the junction of analytic and real-variables. We will explore this junction, from the point of view of Haar shifts and commutators. 1. Haar Functions We consider operators which satisfy invariance properties with respect to two well-known groups. The first group we take to the translation operators (1.1) Try f (x) := f (x − y) , y ∈ R . Note that formally, the adjoint ope...

S. Banach pointed out that the graph of generic (in sense Baire category) element $\text{Homeo}([0,1])$ has length $2$. J. Mycielski asked if measure theoretic dual holds, i.e., all but Haar null many Christensen) elements have We answer this question in affirmative. call $f \in \text{Homeo}([0,1]^d)$ singular it takes a suitable set full to nullset, and strongly is almost everywhere differenti...

We consider compact group generalizations T (n) of the Thue-Morse sequence and prove that the subsequence T (n) is uniformly distributed with respect to a measure ν that is absolutely continuous with respect to the Haar measure. The proof is based on a proper generalization of the Fourier based method of Mauduit and Rivat in their study of the sum-of-digits function of squares to group represen...