Search

Let k \ge 1 be an integer and f a holomorphic endomorphism of \mathbb {P}^k (\mathbb {C}) of algebraic degree d\geq 2. We introduce a dynamical volume dimension for ergodic f-invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than (k-1)\log d, a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k=1, but depends on the dynamics of f to incorporate the absence of an analogue of Koebe's theorem and the non-conformality of holomorphic endomorphisms for k\geq 2. If \nu is an ergodic f-invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of \nu. As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k\geq 1. Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents. [ABSTRACT FROM AUTHOR]

Let \mu and \nu be probability measures in the complex plane, and let p and q be independent random polynomials of degree n, whose roots are chosen independently from \mu and \nu, respectively. Under assumptions on the measures \mu and \nu, the limiting distribution for the zeros of the sum p+q was computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 124719] as n \to \infty. In this paper, we generalize and extend this result to the case where p and q have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of \mu and \nu, scaled by the limiting ratio of the degrees of p and q. Additionally, our approach provides a complete description of the limiting distribution for the zeros of p + q for any pair of measures \mu and \nu, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment. [ABSTRACT FROM AUTHOR]

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature \mathrm {Ric}_N is bounded from below by a real number K in every timelike direction satisfies the timelike curvature-dimension condition \mathrm {TCD}_q(K,N) for all q\in (0,1). The converse and a nonpositive-dimensional version (N \le 0) of this result are also shown. Our discussion is based on the solvability of the Monge problem with respect to the q-Lorentz–Wasserstein distance as well as the characterization of q-geodesics of probability measures. One consequence of our work is the sharp timelike Brunn–Minkowski inequality in the Lorentz–Finsler case. [ABSTRACT FROM AUTHOR]

Almost finiteness was introduced in the seminal work of Kerr [J. Eur. Math. Soc. (JEMS) 22 (2020), pp. 3697–3745] as an dynamical analogue of \mathcal {Z}-stability in the Toms-Winter conjecture. In this article, we provide the first examples of minimal, topologically free actions of amenable groups that have mean dimension zero but are not almost finite. More precisely, we prove that there exists an infinite family of amenable wreath products that admit topologically free, minimal profinite actions on the Cantor space which fail to be almost finite. Furthermore, these actions have dynamical comparison. This intriguing new phenomenon shows that Kerr's dynamical analogue of Toms-Winter conjecture fails for minimal, topologically free actions of amenable groups. The notion of allosteric group holds a significant position in our study. A group is allosteric if it admits a minimal action on a compact space with an invariant ergodic probability measure that is topologically free but not essentially free. We study allostery of wreath products and provide the first examples of allosteric amenable groups. [ABSTRACT FROM AUTHOR]

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with continuous density, denoted by \Lambda ^n: for \nu a finite, even Borel measure on the unit sphere and even \mu \in \Lambda ^n, there exists a symmetric convex body K such that \begin{equation*} d\nu (u)=c_{\mu,K}dS^{\mu }_{K}(u), \end{equation*} where c_{\mu,K} is a quantity that depends on \mu and K and dS^{\mu }_{K}(u) is the surface area-measure of K with respect to \mu. Examples of measures in \Lambda ^n are homogeneous measures (with c_{\mu,K}=1) and probability measures with radially decreasing densities (e.g. the Gaussian measure). We will also consider weighted projection bodies \Pi _\mu K by classifying them and studying the isomorphic Shephard problem: if \mu and \nu are even, homogeneous measures with density and K and L are symmetric convex bodies such that \Pi _{\mu } K \subset \Pi _{\nu } L, then can one find an optimal quantity \mathcal {A}>0 such that \mu (K)\leq \mathcal {A}\nu (L)? Among other things, we show that, in the case where \mu =\nu and L is a projection body, \mathcal {A}=1. [ABSTRACT FROM AUTHOR]

We show that non-trivial chain recurrent classes for generic C^1 star flows satisfy a dichotomy: either they have zero topological entropy, or they must be isolated. Moreover, chain recurrent classes for generic star flows with zero entropy must be sectional hyperbolic, and cannot be detected by any non-trivial ergodic invariant probability measure. As a result, we show that C^1 generic star flows have only finitely many Lyapunov stable chain recurrent classes. [ABSTRACT FROM AUTHOR]

Let \Gamma be a countable discrete group and let H be a lcsc totally disconnected group, L a compact open subgroup of H, and \rho : \Gamma \rightarrow H a homomorphism with dense image. In this paper we construct, for every bi-L-invariant probability measure \theta on H, an explicit Furstenberg discretization \tau of \theta such that the Poisson boundary (B_\theta,\nu _\theta) of (H,\theta) is a \tau-boundary, where \Gamma acts on B_\theta via the homomorphism \rho. We also provide several criteria for when this \tau-boundary is maximal. Our technique can for instance be used to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not L^p-irreducible for any p \geq 1, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we provide the first example of a countable discrete group \Gamma and two spread-out probability measures \tau _1 and \tau _2 on \Gamma such that the boundary entropy spectrum of (\Gamma,\tau _1) is an interval, while the boundary entropy spectrum of (\Gamma,\tau _2) is a Cantor set. [ABSTRACT FROM AUTHOR]

We show that for any even log-concave probability measure \mu on \mathbb {R}^n, any pair of symmetric convex sets K and L, and any \lambda \in [0,1], \begin{equation*} \mu ((1-\lambda) K+\lambda L)^{c_n}\geq (1-\lambda) \mu (K)^{c_n}+\lambda \mu (L)^{c_n}, \end{equation*} where c_n\geq n^{-4-o(1)}. This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Richard J. Gardner and Artem Zvavitch [Tran. Amer. Math. Soc. 362 (2010), pp. 5333–5353]; Andrea Colesanti, Galyna V. Livshyts, Arnaud Marsiglietti [J. Funct. Anal. 273 (2017), pp. 1120–1139]). Moreover, our bound improves for various special classes of log-concave measures. [ABSTRACT FROM AUTHOR]

We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension of order r>0 for an arbitrary compactly supported Borel probability measure \nu is given by its Rényi dimension at the point q_{r} where the L^{q}-spectrum of \nu and the line through the origin with slope r intersect. In particular, this proves the continuity of r\mapsto \overline {D}_{r}(\text {\nu)} as conjectured by Lindsay [ Quantization dimension for probability distributions , ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of North Texas]. This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the L^{q}-spectrum restricted to \left (0,1\right). We give sufficient conditions in terms of the L^{q}-spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a \mathcal {C}^{1}-self-conformal iterated function system on \mathbb {R}^{d} without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved. [ABSTRACT FROM AUTHOR]

We show that the "stabilization" of any countable ergodic probability measure preserving (p.m.p.) equivalence relation which is not Schmidt, i.e. admits no central sequences in its full group, always gives rise to a stable equivalence relation with a unique stable decomposition, providing the first non-strongly ergodic such examples. In the proof, we moreover establish a new local characterization of the Schmidt property. We also prove some new structural results for product equivalence relations and orbit equivalence relations of diagonal product actions. [ABSTRACT FROM AUTHOR]

Let \Gamma be a sofic group, \Sigma be a sofic approximation sequence of \Gamma and X be a \Gamma-subshift with non-negative sofic topological entropy with respect to \Sigma. Further assume that X is a shift of finite type, or more generally, that X satisfies the topological Markov property. We show that for any sufficiently regular potential f \colon X \to \mathbb {R}, any translation-invariant Borel probability measure on X which maximizes the measure-theoretic sofic pressure of f with respect to \Sigma is a Gibbs state with respect to f. This extends a classical theorem of Lanford and Ruelle, as well as previous generalizations of Moulin Ollagnier, Pinchon, Tempelman and others, to the case where the group is sofic. As applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense. On the expository side, we present a short proof of Chung's variational principle for sofic topological pressure. [ABSTRACT FROM AUTHOR]

Neretin in [Izv. Ross. Akad. Nauk Ser. Mat. 77 (2013), pp. 95–108] constructed an analogue of the Hua measures on the infinite p-adic matrices \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right). Bufetov and Qiu in [Compos. Math. 153 (2017), pp. 2482–2533] classified the ergodic measures on \mathrm {Mat}\left (\mathbb {N},\mathbb {Q}_p\right) that are invariant under the natural action of \mathrm {GL}(\infty,\mathbb {Z}_p)\times \mathrm {GL}(\infty,\mathbb {Z}_p). In this paper we solve the problem of ergodic decomposition for the p-adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains. [ABSTRACT FROM AUTHOR]

Which combinatorial sequences correspond to moments of probability measures on the real line? We present a generating function, in the form of a continued fraction, for a fourteen-parameter family of such sequences and interpret these in terms of combinatorial statistics on the symmetric groups. Special cases include several classical and noncommutative probability laws, along with a substantial subset of the orthogonalizing measures in the q-Askey scheme, now given a new combinatorial interpretation in terms of elementary permutation statistics. This framework further captures a variety of interesting combinatorial sequences including, notably, the moment sequences associated to distributions of the numbers of occurrences of (classical and vincular) permutation patterns of length three. This connection between pattern avoidance and broader ideas in classical and noncommutative probability is among several intriguing new corollaries, which generalize and unify results previously appearing in the literature, while opening up new lines of inquiry. The fourteen combinatorial statistics further generalize to signed and colored permutations, and, as an infinite family of statistics, to the k-arrangements : permutations with k-colored fixed points, introduced here along with several related results and conjectures. [ABSTRACT FROM AUTHOR]

It is natural to ask whether it is possible to construct Lévy-like processes where actions by random elements of a given semigroup play the role of increments. Such semigroups induce a convolution-like algebra structure in the space of finite measures. In this paper, we show that the Whittaker convolution operator, related with the Shiryaev process, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. We then introduce the class of Lévy processes with respect to the Whittaker convolution and study their basic properties. We obtain a martingale characterization of the Shiryaev process analogous to Lévy's characterization of Brownian motion. Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed for differential operators whose associated convolution does not satisfy the usual compactness assumption on the support of the convolution. [ABSTRACT FROM AUTHOR]

Let μ be an α-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform μ. More precisely, if H is a truncated hyperbolic paraboloid in Rd we study the optimal β for which ∫H |μ(R\xi)|2, d σ (ξ) ≤ C(α, μ) R−β for all R > 1. Our estimates for β depend on the minimum between the number of positive and negative principal curvatures of H; if this number is as large as possible our estimates are sharp in all dimensions. [ABSTRACT FROM AUTHOR]

We study the measurable dynamics of a family of skew-product maps known as Viana maps. In our setting, those maps are constructed by coupling two quadratic maps, and we aim at showing the abundance of non-uniform hyperbolicity in this family. We prove that, for any polynomial coupling function of odd degree, when the parameter pair of the two factor quadratic maps is chosen from a two-dimensional positive measure set, the associated Viana map has two positive Lyapunov exponents and admit finitely many ergodic absolutely continuous invariant probability measures. [ABSTRACT FROM AUTHOR]

Let μ be a probability measure on R/Z that is ergodic under the × p map, with positive entropy. In 1995, Host showed that if gcd(m,p) = 1, then μ almost every point is normal in base m. In 2001, Lindenstrauss showed that the conclusion holds under the weaker assumption that p does not divide any power of m. In 2015, Hochman and Shmerkin showed that this holds in the ''correct'' generality, i.e., if m and p are independent. We prove a simultaneous version of this result: for μ typical x, if m > p are independent, we show that the orbit of (x,x) under (× m, × p) equidistributes for the product of the Lebesgue measure with \μ. We also show that if m > n > 1 and n is independent of p as well, then the orbit of (x,x) under (× m, × n) equidistributes for the Lebesgue measure. [ABSTRACT FROM AUTHOR]

Let Γ be a lattice of a semisimple Lie group L. Suppose that a one parameter Ad-diagonalizable subgroup gt of L acts ergodically on L/Γ with respect to the probability Haar measure μ. For certain proper subgroup U of the unstable horospherical subgroup of gt and certain x ∈ L/Γ we show that for almost every u ∈ U the trajectory gtux: 0 ≤ t ≤ T is equidistributed with respect to μ as T → ∞. [ABSTRACT FROM AUTHOR]

In this work we prove a universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space X. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of random holomorphic sections is independent of the choice of the probability measure on the space of holomorphic sections. In the case when X is a compact Kähler manifold, we also prove an off-diagonal exponential decay estimate for the Bergman kernels of a sequence of positive line bundles on X. [ABSTRACT FROM AUTHOR]

We analyze the rank gradient of finitely generated groups with respect to sequences of subgroups of finite index that do not necessarily form a chain, by connecting it to the cost of p.m.p. (probability measure preserving) actions. We generalize several results that were only known for chains before. The connection is made by the notion of local-global convergence. In particular, we show that for a finitely generated group Γ with fixed price c, every Farber sequence has rank gradient c-1. By adapting Lackenby's trichotomy theorem to this setting, we also show that in a finitely presented amenable group, every sequence of subgroups with index tending to infinity has vanishing rank gradient. [ABSTRACT FROM AUTHOR]

Let α : G X be a continuous action of an infinite countable group on a compact Hausdorff space. We show that, under the hypothesis that the action a is topologically free and has no G-invariant regular Borel probability measure on X, dynamical comparison implies that the reduced crossed product of a is purely infinite and simple. This result, as an application, shows a dichotomy between stable finiteness and pure infiniteness for reduced crossed products arising from actions satisfying dynamical comparison. We also introduce the concepts of paradoxical comparison and the uniform tower property. Under the hypothesis that the action a is exact and essentially free, we show that paradoxical comparison together with the uniform tower property implies that the reduced crossed product of a is purely infinite. As applications, we provide new results on pure infiniteness of reduced crossed products in which the underlying spaces are not necessarily zero dimensional. Finally, we study the type semigroups of actions on the Cantor set in order to establish the equivalence of almost unperforation of the type semigroup and comparison. This sheds light on a question arising in a paper of Rørdam and Sierakowski. [ABSTRACT FROM AUTHOR]

Let Γn be an n × n Haar-invariant orthogonal matrix. Let Zn be the p × q upper-left submatrix of Γn, where p = pn and q = qn are two positive integers. Let Gn be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between √ nZn and Gn in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance, and the Euclidean distance. We prove that each of the first three distances goes to zero as long as pq/n goes to zero, and not so if (p, q) sits on the curve pq = σn, where σ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq²/n goes to zero, and not so if (p, q) sits on the curve pq² = σn. A previous work by Jiang (2006) shows that the total variation distance goes to zero if both p/ √ n and q/ √ n go to zero, and it is not true provided p = c √ n and q = d √ n with c and d being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as pq/n → 0 and the distance does not go to zero if pq = σn for some constant σ. [ABSTRACT FROM AUTHOR]

We propose a generalization of the Poincaré-Birkhoff Theorem on area-preserving twist maps to area-preserving twist maps F that are random with respect to an ergodic probability measure. In this direction, we will prove several theorems concerning existence, density, and type of the fixed points. To this end first we introduce a randomized version of generalized generating functions, and verify the correspondence between its critical points and the fixed points of F, a fact which we successively exploit in order to prove the theorems. The study we carry out needs to combine probabilistic techniques with methods from nonlinear PDE, and from differential geometry, notably Moser’s method and Conley-Zehnder theory. Our stochastic model in the periodic case coincides with the classical setting of the Poincaré-Birkhoff Theorem. [ABSTRACT FROM AUTHOR]

A metric measure space is a complete, separable metric space equipped with a probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov-Prohorov metric of Greven, Pfaffelhuber and Winter. We consider the natural binary operation ⊞ on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup with a translation invariant metric. There is an explicit family of continuous semicharacters that is extremely useful for, inter alia, establishing that there are no infinitely divisible elements and that each element has a unique factorization into prime elements. We investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric. For example, we show that for any given positive real numbers a, b, c the trivial space is the only space X that satisfies aX ⊞ bX = cX. We establish that there is no analogue of the law of large numbers: if X1,X2, . . . is an identically distributed independent sequence of random spaces, then no subsequence of 1/n ⊞nk=1 Xk converges in distribution unless each Xk is almost surely equal to the trivial space. We characterize the infinitely divisible probability measures and the Lévy processes on this semigroup, characterize the stable probability measures and establish a counterpart of the LePage representation for the latter class. [ABSTRACT FROM AUTHOR]

In this paper, we show that each expanding Thurston map f: S²→ S² has 1+deg f fixed points, counted with appropriate weight, where deg f denotes the topological degree of the map f. We then prove the equidistribution of preimages and of (pre)periodic points with respect to the unique measure of maximal entropy μf for f. We also show that (S², f,μf) is a factor of the left shift on the set of one-sided infinite sequences with its measure of maximal entropy, in the category of measure-preserving dynamical systems. Finally, we prove that μf is almost surely the weak* limit of atomic probability measures supported on a random backward orbit of an arbitrary point. [ABSTRACT FROM AUTHOR]

For the generic continuous map and for the generic homeomorphism of the Cantor space, we study the dynamics of the induced map on the space of probability measures, with emphasis on the notions of Li-Yorke chaos, topological entropy, equicontinuity, chain continuity, chain mixing, shadowing and recurrence. We also establish some results concerning induced maps that hold on arbitrary compact metric spaces. [ABSTRACT FROM AUTHOR]

We define positively expansive and expansive measures on uniform spaces extending the analogous concepts on metric spaces. We show that such measures can exist for measurable or bimeasurable maps on compact non-Hausdorff uniform spaces. We prove that positively expansive probability measures on Lindelöf spaces are non-atomic and their corresponding maps eventually aperiodic. We prove that the stable classes of measurable maps have measure zero with respect to any positively expansive invariant measure. In addition, any measurable set where a measurable map in a Lindelöf uniform space is Lyapunov stable has measure zero with respect to any positively expansive inner regular measure. We conclude that the set of sinks of any bimeasurable map with canonical coordinates of a Lindelöf space has zero measure with respect to any positively expansive inner regular measure. Finally, we show that every measurable subset of points with converging semiorbits of a bimeasurable map on a separable uniform space has zero measure with respect to every expansive outer regular measure. These results generalize those found in works by Arbieto and Morales and by Reddy and Robertson. [ABSTRACT FROM AUTHOR]

The equivalences to and the connections between the modulus-of-continuity condition, compact containment and tightness on DE[a, b] with a < b are studied. The results within are tools for establishing tightness for probability measures on DE[a, b] that generalize and simplify prevailing results in the cases that E is a metric space, nuclear space dual or, more generally, a completely regular topological space. Applications include establishing weak convergence to martingale problems, the long-time typical behavior of nonlinear filters and particle approximation of cadlag probability-measure-valued processes. This particle approximation is studied herein, where the distribution of the particles is the underlying measure-valued process at an arbitrarily fine discrete mesh of points. [ABSTRACT FROM AUTHOR]

For discrete measured groupoids preserving a probability measure we introduce a notion of sofic dimension that measures the asymptotic growth of the number of sofic approximations on larger and larger finite sets. In the case of groups we give a formula for free products with amalgamation over an amenable subgroup. We also prove a free product formula for measurepreserving actions. [ABSTRACT FROM AUTHOR]

Ergodic properties of rational maps are studied, generalising the work of F. Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for an ergodic invariant probability measure with positive Lyapunov exponent to be absolutely continuous with respect to a general conformal measure. If they hold, we can construct an induced expanding Markov map with integrable return time which generates the invariant measure. [ABSTRACT FROM AUTHOR]

We develop cycle index generating functions for orthogonal groups in even characteristic and give some enumerative applications. A key step is the determination of the values of the complex linear-Weil characters of the finite symplectic group, and their induction to the general linear group, at unipotent elements. We also define and study several natural probability measures on integer partitions. [ABSTRACT FROM AUTHOR]

The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen (2007) is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in Rd and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized Baumslag-Solitar group. In contrast, the fiber measures for filters on inflated fractal spaces cannot be atomic, and thus can support neither MSF wavelets nor systems of imprimitivity. [ABSTRACT FROM AUTHOR]

Assume that $ \Omega\subset\mathbb{C}^{n}$ boundary). We denote $ \left\Vert f\right\Vert _{z}^{2}:=\int_{0}^{1}\vert f(e^{2\pi it}z)\vert^{2}dt$0$ --> $ \varepsilon>0$ be a circular invariant Borel probability measure on $ \partial\Omega$ $ g\in A(\Omega)$ is a continuous function on $ \partial\Omega$ on $ \partial\Omega$ $ f_{1},f_{2}\in A(\Omega)$ $ \left\Vert g+f_{1}\right\Vert _{z}\leq\left\Vert h\right\Vert _{z}$ $ \vert(g+f_{2})(z)\vert\leq\max_{\vert\lambda\vert=1}h(\lambda z)$ $ z\in\partial\Omega$

We study the Steiner problem of finding a minimal spanning network in the setting of a space of probability measures with metric defined by the cost of optimal transport between measures. The existence of a solution is shown for the Wasserstein space $ P_p(\mathcal{X})$ $ \mathcal{X}$ $ P_2(\mathbb{R}^n)$ [ABSTRACT FROM AUTHOR]

Let $G$ be a locally compact group. Blum and Eisenberg proved that if $G$ is abelian, then a sequence of probability measures on $G$ is strongly ergodic if and only if the sequence converges weakly to the Haar measure on the Bohr compactification of $G.$ In this paper, we shall prove an extension of Blum and Eisenberg's Theorem for ergodic sequences in the Fourier-Stieltjes algebra of $G.$ We shall also give an improvement to Milnes and Paterson's more recent generalization of Blum and Eisenberg's result to general locally compact groups, and we answer a question of theirs on the existence of strongly (or weakly) ergodic sequences of measures on $G.$ [ABSTRACT FROM AUTHOR]

Let $X_1,X_2,\dotsc$ be a sequence of i.i.d. random variables each taking values of 1 and $-1$ with equal probability. For $1/2<\rho<1$ satisfying the equation $1-\rho-\dotsb-\rho^s=0$, let $\mu$ be the probability measure induced by $S=\sum_{i=1}^\infty \rho^iX_i$. For any $x$ in the range of $S$, let \[d(\mu,x)=\lim_{r\to0^+}\log\mu([x-r,x+r])/\log r] be the local dimension of $\mu$ at $x$ whenever the limit exists. We prove that \[\alpha^*=-\frac{\log 2}{\log\rho}\quad\text{and}\quad \alpha_*=-\frac{\log\de}{s\log\rho}-\frac{\log 2}{\log\rho},] where $\delta=(\sqrt{5}-1)/2$, are respectively the maximum and minimum values of the local dimensions. If $s=2$, then $\rho$ is the golden number, and the approximate numerical values are $\alpha^*\approx 1.4404$ and $\alpha_*\approx0.9404$. [ABSTRACT FROM AUTHOR]