Packing measure of super separated iterated function systems

Let J be the limit set of an iterated function system in \(\mathbb {R}^d\) satisfying the open set condition. It is well known that the h-dimensional packing measure of J is positive and finite when h is given by Hutchinson’s formula. However, it may be hard to find a formula for the h-dimensional packing measure of J. We introduce the super separation condition and use it to reduce the problem of computing the packing measure to checking densities of a finite number of balls around each point in the limit set. We then use this fact to find formulas for the packing measure of a class of Cantor sets in \(\mathbb {R}\), a class of fractals based on regular convex polygons in \(\mathbb {R}^2\), and a class of fractals based on regular simplexes in \(\mathbb {R}^d\) for \(d \ge 3\).     

Projections of Patterson-Sullivan measures and the dichotomy of Mohammadi-Oh

Let Γ be some discrete subgroup of SO°(n + 1, R) with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the N component in some Iwasawa decomposition SO°(n+1, R) = KAN. We also study the dimension of projected Patterson-Sullivan measures along some fixed direction.     

Quadratic differentials (A(z − a)(z − b)/(z − c)2) dz2 and algebraic Cauchy transform

We discuss the representability almost everywhere (a.e.) in C of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials A(z ? a)(z ? b)×(z ? c)^?2 dz². More precisely, we give a necessary and sufficient condition on the complex numbers a and b for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.     

A Federer-style characterization of sets of finite perimeter on metric spaces

In the setting of a metric space equipped with a doubling measure that supports a Poincaré inequality, we show that a set E is of finite perimeter if and only if \({\mathcal {H}}(\partial ^1 I_E)<\infty \), that is, if and only if the codimension one Hausdorff measure of the 1-fine boundary of the set’s measure theoretic interior \(I_E\) is finite. To obtain the necessity of the above condition, we prove a suitable characterization of the 1-fine boundary, analogously to what is known in the case \(p>1\), and apply a quasicontinuity-type result for \(\mathrm {BV}\) functions proved in the metric setting by Lahti and Shanmugalingam (J Math Pures Appl (9) 107(2):150–182, 2017). To obtain the sufficiency, we generalize further results of fine potential theory from the case \(p>1\) to the case \(p=1\), including weak analogs of a Cartan property for solutions of obstacle problems, and of the Choquet property for finely open sets.     

A sharp bound on the Hausdorff dimension of the singular set of a uniform measure

The study of the geometry of n-uniform measures in \(\mathbb {R}^{d}\) has been an important question in many fields of analysis since Preiss’ seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is only one known example of a non-trivial uniform measure, namely 3-Hausdorff measure restricted to the Kowalski–Preiss cone. Using this cone one can construct an n-uniform measure whose singular set has Hausdorff dimension \(n-3\). In this paper, we prove that this is the largest the singular set can be. Namely, the Hausdorff dimension of the singular set of any n-uniform measure is at most \(n-3\).     

Multi-invariant measures and subsets on nilmanifolds

Given a Z^r-action α on a nilmanifold X by automorphisms and an ergodic α-invariant probability measure μ, we show that μ is the uniform measure on X unless, modulo finite index modification, one of the following obstructions occurs for an algebraic factor action

1. (1)
   the factor measure has zero entropy under every element of the action
    
2. (2)
   the factor action is virtually cyclic.
    

We also deduce a rigidity property for invariant closed subsets.

     

Nodal sets of smooth functions with finite vanishing order and <Emphasis Type="Italic">p</Emphasis>-sweepouts

We show that on a compact Riemannian manifold (M, g), nodal sets of linear combinations of any \(p+1\) smooth functions form an admissible p-sweepout provided these linear combinations have uniformly bounded vanishing order. This applies in particular to finite linear combinations of Laplace eigenfunctions. As a result, we obtain a new proof of the Gromov, Guth, Marques–Neves upper bounds on the min–max p-widths of M. We also prove that close to a point at which a smooth function on \(\mathbb {R}^{n+1}\) vanishes to order k, its nodal set is contained in the union of \(k\,W^{1,p}\) graphs for some \(p > 1\). This implies that the nodal set is locally countably n-rectifiable and has locally finite \(\mathcal {H}^n\) measure, facts which also follow from a previous result of Bär. Finally, we prove the continuity of the Hausdorff measure of nodal sets under heat flow.     

Minimality of Convergence in Measure Topologies on Finite von Neumann Algebras

We prove that the natural embedding of the metric ideal space on a finite von Neumann algebra $\mathcal{M}$ into the *-algebra of measurable operators $\tilde {\mathcal {M}}$ endowed with the topology of convergence in measure is continuous. Using this fact, we prove that the topology of convergence in measure is a minimal one among all metrizable topologies consistent with the ring structure on $\tilde {\mathcal {M}}$ .     

On admissible strategies in robust utility maximization

The existence of optimal strategy in robust utility maximization is addressed when the utility function is finite on the entire real line. A delicate problem in this case is to find a ??good definition?? of admissible strategies to admit an optimizer. Under certain assumptions, especially a kind of time-consistency property of the set ${\mathcal{P}}$ of probabilities which describes the model uncertainty, we show that an optimal strategy is obtained in the class of those whose wealths are supermartingales under all local martingale measures having a finite generalized entropy with one of ${P\in\mathcal{P}}$ .     

Support Recovery for Sparse Super-Resolution of Positive Measures

We study sparse spikes super-resolution over the space of Radon measures on \(\mathbb {R}\) or \(\mathbb {T}\) when the input measure is a finite sum of positive Dirac masses using the BLASSO convex program. We focus on the recovery properties of the support and the amplitudes of the initial measure in the presence of noise as a function of the minimum separation t of the input measure (the minimum distance between two spikes). We show that when \({w}/\lambda \), \({w}/t^{2N-1}\) and \(\lambda /t^{2N-1}\) are small enough (where \(\lambda \) is the regularization parameter, w the noise and N the number of spikes), which corresponds roughly to a sufficient signal-to-noise ratio and a noise level small enough with respect to the minimum separation, there exists a unique solution to the BLASSO program with exactly the same number of spikes as the original measure. We show that the amplitudes and positions of the spikes of the solution both converge toward those of the input measure when the noise and the regularization parameter drops to zero faster than \(t^{2N-1}\).     

Two results for monocompact measures

For every finite measure space (Ω,A, P) whereA is K₁-generated we prove the equivalence of compactness and monocompactness for P . Moreover, we prove the existence of a perfect, not monocompaot probability, thus answering an open question in [6]. Let P be a charge on the algebraA andK ?A be a monocompact class. We show that P is o-additive ifK ^S P-approximatesK ^S, the family of finite unions inK , needs not to be monocompact.     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

Spectral self-affine measures with prime determinant

The self-affine measure $\mu _{M,D}$ relating to an expanding matrix $M\in M_{n}(\mathbb Z )$ and a finite digit set $D\subset \mathbb Z ^n$ is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of $\mu _{M,D}$ in the case when $|\det (M)|=p$ is a prime. The main result shows that under certain mild conditions, if there are two points $s_{1}, s_{2}\in \mathbb R ^{n}, s_{1}-s_{2}\in \mathbb Z ^{n}$ such that the exponential functions $e_{s_{1}}(x), e_{s_{2}}(x)$ are orthogonal in $L^{2}(\mu _{M,D})$ , then the self-affine measure $\mu _{M,D}$ is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure.     

On sets of directions determined by subsets of ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Given E ? ? ^d , d ≥ 2, define

$D(E) \equiv \left\{ {{{x - y} \over {\left| {x - y} \right|}}:x,y \in E} \right\} \subset {S^{d - 1}}$

to be the set of directions determined by E. We prove that if the Hausdorff dimension of E is greater than d ? 1, then σ(D(E)) > 0, where σ denotes the surface measure on S ^d?1. In the process, we prove some tight upper and lower bounds for the maximal function associated with the Radon-Nikodym derivative of the natural measure on D. This result is sharp, since the conclusion fails to hold if E is a (d ? 1)-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir ([22, 23]) on directions determined by finite subsets of ? ^d . We also discuss the case when the Hausdorff dimension of E is precisely d ? 1, where some interesting counter-examples have been obtained by Simon and Solomyak ([25]) in the planar case. In response to the conjecture stated in this paper, T. Orponen and T. Sahlsten ([20]) have recently proved that if the Hausdorff dimension of E equals d ? 1 and E is rectifiable and is not contained in a hyper-pane, the Lebesgue measure of the set of directions is still positive. Finally, we show that our continuous results can be used to recover and, in some cases, improve the exponents for the corresponding results in the discrete setting for large classes of finite point sets. In particular, we prove that a finite point set P ? ? ^d , d ≥ 3, satisfying a certain discrete energy condition (Definition 3.1) determines ? #P distinct directions.

     

Generalized Lebesgue points for Sobolev functions

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point x in a metric measure space (X, d, μ) is called a generalized Lebesgue point of a measurable function f if the medians of f over the balls B(x, r) converge to f(x) when r converges to 0. We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function. We show that a function f ∈ M ^s,p (X), 0 < s ≤ 1, 0 < p < 1, where X is a doubling metric measure space, has generalized Lebesgue points outside a set of \(\mathcal{H}^h\)-Hausdorff measure zero for a suitable gauge function h.     

VC-sets and generic compact domination

Let X be a closed subset of a locally compact second countable group G whose family of translates has finite VC-dimension. We show that the topological border of X has Haar measure 0. Under an extra technical hypothesis, this also holds if X is constructible. We deduce from this generic compact domination for definably amenable NIP groups.     

A ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolic three-space

A group $G$ is said to be periodic if for every $g\in G$ there exists a positive integer $n$ with $g^n=\mathrm{Id}$ . We prove that a finitely generated periodic group of homeomorphisms on the 2-torus that preserves a probability measure $\mu $ is finite. Moreover if the group consists of homeomorphisms isotopic to the identity, then it is abelian and acts freely on $\mathbb{T }^2$ . In the Appendix, we show that every finitely generated 2-group of toral homeomorphisms is finite.     

Synchronization and random long time dynamics for mean-field plane rotators

We consider the natural Langevin dynamics which is reversible with respect to the mean-field plane rotator (or classical spin XY) measure. It is well known that this model exhibits a phase transition at a critical value of the interaction strength parameter \(K\) , in the limit of the number \(N\) of rotators going to infinity. A Fokker–Planck PDE captures the evolution of the empirical measure of the system as \(N \rightarrow \infty \) , at least for finite times and when the empirical measure of the system at time zero satisfies a law of large numbers. The phase transition is reflected in the fact that the PDE for \(K\) above the critical value has several stationary solutions, notably a stable manifold—in fact, a circle—of stationary solutions that are equivalent up to rotations. These stationary solutions are actually unimodal densities parametrized by the position of their maximum (the synchronization phase or center). We characterize the dynamics on times of order \(N\) and we show substantial deviations from the behavior of the solutions of the PDE. In fact, if the empirical measure at time zero converges as \(N \rightarrow \infty \) to a probability measure (which is away from a thin set that we characterize) and if time is speeded up by \(N\) , the empirical measure reaches almost instantaneously a small neighborhood of the stable manifold, to which it then sticks and on which a non-trivial random dynamics takes place. In fact the synchronization center performs a Brownian motion with a diffusion coefficient that we compute. Our approach therefore provides, for one of the basic statistical mechanics systems with continuum symmetry, a detailed characterization of the macroscopic deviations from the large scale limit—or law of large numbers—due to finite size effects. But the interest for this model goes beyond statistical mechanics, since it plays a central role in a variety of scientific domains in which one aims at understanding synchronization phenomena.     

Logarithmic and Linear Potentials of Signed Measures and Markov Property of Associated Gaussian Fields

We consider the family of finite signed measures on the complex plane \(\mathbb {C}\) with compact support, of finite logarithmic energy and with zero total mass. We show directly that the logarithmic potential of such a measure sits in the Beppo Levi space, namely, the extended Dirichlet space of the Sobolev space of order 1 over \(\mathbb {C}\), and that the half of its Dirichlet integral equals the logarithmic energy of the measure. We then derive the (local) Markov property of the Gaussian field \(\textbf {G}(\mathbb {C})\) indexed by this family of measures. Exactly analogous considerations will be made for the Beppo Levi space over the upper half plane \(\mathbb {H}\) and the Cameron-Martin space over the real line \(\mathbb {R}\). Some Gaussian fields appearing in recent literatures related to mathematical physics will be interpreted in terms of the present field \(\textbf {G}(\mathbb {C})\).     

The Homomorphism Lattice Induced by a Finite Algebra

Each finite algebra A induces a lattice L _A via the quasi-order → on the finite members of the variety generated by A, where B →C if there exists a homomorphism from B to C. In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice L _A induced by a finite algebra A?’ Our main result is that each finite distributive lattice arises as L _Q , for some quasi-primal algebra Q. We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form L ⊕1, where L is an interval in the subgroup lattice of a finite group.