Möbius disjointness for topological models of ergodic measure-preserving systems with quasi-discrete spectrum

By using a decomposition result for ergodic measure-preserving system with quasi-discrete spectrum, we prove that a generic point of an ergodic quasi-discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in Sarnak's Möbius Disjointness Conjecture. As a direct application, we have that Sarnak's Möbius Disjointness Conjecture holds for any topological model of an ergodic measure-preserving system with quasi-discrete spectrum.     

On Möbius form and Möbius isoparametric hypersurfaces

An umbilic-free hypersurface in the unit sphere is called MSbius isoparametric if it satisfies two conditions, namely, it has vanishing MSbius form and has constant MSbius principal curvatures. In this paper, under the condition of having constant MSbius principal curvatures, we show that the hypersurface is of vanishing MSbius form if and only if its MSbius form is parallel with respect to the Levi-Civita connection of its MSbius metric. Moreover, typical examples are constructed to show that the condition of having constant MSbius principal curvatures and that of having vanishing MSbius form are independent of each other.     

The Problem of Freeness for Euler Monoids and Möbius Groups

The Euler monoid E_n = {(a,b,t) epsilon Z³ : a² + b² = tⁿ, n S 1, is free if and only if n is odd (Theorem 1). We extend the results of Lyndon and Ullman, and Beardon concerning the set of those rational numbers mu epsilon (-2,2) for which the matrix Möbius group G_mu generated by A= and B = is not free (Theorems 2, 3, 4).     

Möbius and triangle semigroups

The purpose of this paper is to describe the structures of the Möbius semigroup induced by the Möbius transformation group (?, SL(2,?)). In particular, we study stabilizer subsemigoups of Möbius semigroup via the triangle semigroup. In this work, we obtained a geometric interpretation of the least contraction coefficient function of the Möbius semigroup via the triangle semigroup and investigated an extension of stabilizer subsemigoups of the Möbius semigroup. Finally, we obtained a factorization of our stabilizer subsemigoups of the Möbius semigroup.     

Riemann Sums and Möbius

Let S be the set of square-free natural numbers. A Hilbert-Schmidt operator, A, associated to the Möbius function has the property that it maps from \({ \cup _{0 < r < \infty }}{l^r}(s)\) to \({ \cap _{0 < r < \infty }}{l^r}(s)\), injectively. If 0 < r< 2 and ξ ∈ l^r (S), the series \({f_\zeta } = \sum\nolimits_{n \in s} {A\zeta (x)cos2\pi nx} \) converges uniformly to an element of f_ξR₀, i.e., a periodic, even, continuous function with equally spaced Riemann sums, \(\sum\nolimits_{j = 0}^{N - 1} {{f_\zeta }} (j/N) = 0,N = 1,2....\) If \({A_{\zeta \lambda }} = \lambda {\zeta _\lambda },{\zeta _\lambda }(1) = 1\), then ξ_λ is multiplicative. If \({f_{{\zeta _\lambda }}} \in {\Lambda _a}\), the space of α-Lipschitz continous functions, for some α > 0, and if χ is any Dirichlet character, then L(s, χ) ≠ 0, Res > 1 ? α. Conjecturally, the Generalized Riemann Hypothesis (GRH) is equivalent to f_ξ ∈ Λ_α, α < 1/2, ξ ∈ l^r (S), 0 < r < 2. Using a 1991 estimate by R. C. Baker and G. Harman, one finds GRH implies f_ξ ∈ Λ_α, α < 1/4, ξ ∈ l^r (S), 0 < r < 2. The question of whether R₀ ∩ Λ_α ≠ {0} for some positive α > 0 is open.     

Möbius characterization of hemispheres

In this paper we generalize the Möbius characterization of metric spheres as obtained in Foertsch and Schroeder [4] to a corresponding Möbius characterization of metric hemispheres.     

Factorizations of Möbius Gyrogroups

In this paper we consider a M?bius gyrogroup on a real Hilbert space (of finite or infinite dimension) and we obtain its factorization by gyrosubgroups and subgroups. It is shown that there is a duality relation between the quotient spaces and the orbits obtained. As an example we will present the factorization of the M?bius gyrogroup of the unit ball in \mathbbRⁿ{\mathbb{R}}^{n} linked to the proper Lorentz group Spin⁺(1, n).     

The Möbius midpoint condition as a test for quasiconformality and the quasimöbius property

The Möbius midpoint condition, introduced by Goldberg in 1974 as a criterion for the quasisymmetry of a mapping of the line onto itself and considered by Aseev and Kuzin in 1998 in the same role for the topological embeddings of the line into ? ⁿ , yields no information on the quasiconformality or quasisymmetry of a topological embedding of ? ^k into ? ⁿ for 1 < k ≤ n. In this article we introduce a Möbius-invariant modification of the midpoint condition, which we call the “Möbius midpoint condition” MMC(f) ≤ H < 1. We prove that if this condition is fulfilled then every homeomorphism of domains in \(\overline {\mathbb{R}^n }\) is K(H)-quasiconformal, while a topological embedding of the sphere \(\overline {\mathbb{R}^k }\) into \(\overline {\mathbb{R}^n }\) (for 1 ≤ k ≤ n) is ω _H-quasimöbius. The quasiconformality coefficient of K(H) and the distortion function ω _H depend only on H and are expressed by explicit formulas showing that K(H) → 1 and ω _H → id as H → 1/2. Since MMC(f) = 1/2 is equivalent to the Möbius property of f, the resulting formulas yield the closeness of the mapping to a Möbius mapping for H near 1/2.     

Metric Möbius geometry and a characterization of spheres

We obtain a Möbius characterization of the n-dimensional spheres S ⁿ endowed with the chordal metric d ₀. We show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is Möbius equivalent to (S ⁿ , d ₀) for some n ≥ 1.     

Trees and ultrametric Möbius structures

We define the concept of an ultrametric Möbius space (Z,M) and show that the boundary at infinity of a nonelementary geodesically complete tree is naturally an ultrametric Möbius space. In addition, we construct to a given ultrametric Möbius space (Z,M) a nonelementary geodesically complete tree, unique up to isometry, with (Z,M) being its boundary at infinity. This yields a one-to-one correspondence.     

On Möbius Duality and Coarse-Graining

We study duality relations for zeta and Möbius matrices and monotone conditions on the kernels. We focus on the cases of families of sets and partitions. The conditions for positivity of the dual kernels are stated in terms of the positive Möbius cone of functions, which is described in terms of Sylvester formulae. We study duality under coarse-graining and show that an \(h\)-transform is needed to preserve stochasticity. We give conditions in order that zeta and Möbius matrices admit coarse-graining, and we prove they are satisfied for sets and partitions. This is a source of relevant examples in genetics on the haploid and multi-allelic Cannings models.     

Topological classification of Möbius transformations

Linear fractional transformations on the extended complex plane are classified up to topological conjugacy. Recall that two transformations f and g are called topologically conjugate if there exists a homeomorphism h such that g = h ^?1 ? f ? h, where ? is the composition of mappings.     

The Extension of Möbius Groups

Let G be a non-elementary subgroup of SL(2,Г _n ) containing hyperbolic elements. We show that G is the extension of a subgroup of SL(2,C) if and only if that G is conjugate in SL(2,Г _n ) to a group G' with the following properties: (1) There are g ₀, h ? G', where g ₀ and h are hyperbolic, such that fix(g ₀) = {0,∞}, fix(h)∩fix(g ₀) =  and fix(h) ∩ C ≠ ; (2) tr(g) ? C for each g ? G'. As an application, we show that if G contains only hyperbolic elements and uniformly parabolic elements, then G is the extension of a subgroup of SL(2,C), which also yields the discreteness of G.