A map maintaining the orbits of a given $ℤ^{d}$-action

Bartosz Frej; Agata Kwaśnicka

Displaying similar documents to “A map maintaining the orbits of a given $ℤ^{d}$ -action”

Topological friction in aperiodic minimal $ℝ^{m}$ -actions

Jarosław Kwapisz (2010)

Fundamenta Mathematicae

Similarity:

For a continuous map f preserving orbits of an aperiodic $ℝ^{m}$ -action on a compact space, its displacement function assigns to x the “time” $t \in ℝ^{m}$ it takes to move x to f(x). We show that this function is continuous if the action is minimal. In particular, f is homotopic to the identity along the orbits of the action.

The orbits of the Hurwitz action of the braid groups on the standard generators

Yoshiro Yaguchi (2010)

Fundamenta Mathematicae

Similarity:

The Hurwitz action of the n-braid group Bₙ on the n-fold direct product $(B_{m}) ⁿ$ of the m-braid group $B_{m}$ is studied. We show that the orbit of any n- tuple of the n standard generators of $B_{n + 1}$ consists of the (n-1)th powers of n+1 elements.

Non-orbit equivalent actions of $𝔽_{n}$

Adrian Ioana (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

For any $2 \leq n \leq \infty$ , we construct a concrete 1-parameter family of non-orbit equivalent actions of the free group $𝔽_{n}$ . These actions arise as diagonal products between a generalized Bernoulli action and the action $𝔽_{n} ↷ (𝕋^{2}, λ^{2})$ , where $𝔽_{n}$ is seen as a subgroup of ${SL}_{2} (ℤ)$ .

Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets

Dae San Kim, Hyun Kwang Kim

Similarity:

Let P be a poset on the set [m]×[n], which is given as the disjoint sum of posets on ’columns’ of [m]×[n], and let P̌ be the dual poset of P. Then P is called a generalized Niederreiter-Rosenbloom-Tsfasman poset (gNRTp) if all further posets on columns are weak order posets of the ’same type’. Let G (resp. Ǧ) be the group of all linear automorphisms of the space $_{q}^{m \times n}$ preserving the P-weight (resp. P̌-weight). We define two partitions of $_{q}^{m \times n}$ , one consisting of ’P-orbits’ and the other of...

Diagonal points having dense orbit

T. K. Subrahmonian Moothathu (2010)

Colloquium Mathematicae

Similarity:

Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f \times f ² \times \dots \times f^{m} : X^{m} \to X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f \times f ² \times \dots \times f^{m}$ . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying...

Finite orbit decomposition of real flag manifolds

Bernhard Krötz, Henrik Schlichtkrull (2016)

Journal of the European Mathematical Society

Similarity:

Let $G$ be a connected real semi-simple Lie group and $H$ a closed connected subgroup. Let $P$ be a minimal parabolic subgroup of $G$ . It is shown that $H$ has an open orbit on the flag manifold $G / P$ if and only if it has finitely many orbits on $G / P$ . This confirms a conjecture by T. Matsuki.

On the stratification of the orbit space for the action of automorphisms on connections

Witold Kondracki, Jan Rogulski

Similarity:

CONTENTSIntroduction..................................................................................................................................................5§1. Basic notions and notation.....................................................................................................................7 1.1. Automorphisms of principal bundles....................................................................................................7 1.2. Connections and parallel...

More classes of non-orbit-transitive operators

Carl Pearcy, Lidia Smith (2010)

Studia Mathematica

Similarity:

In [JKP] and its sequel [FPS] the authors initiated a program whose (announced) goal is to eventually show that no operator in ℒ(ℋ) is orbit-transitive. In [JKP] it is shown, for example, that if T ∈ ℒ(ℋ) and the essential (Calkin) norm of T is equal to its essential spectral radius, then no compact perturbation of T is orbit-transitive, and in [FPS] this result is extended to say that no element of this same class of operators is weakly orbit-transitive. In the present note we show...

Invariant symbolic calculus for semidirect products

Benjamin Cahen (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $G$ be the semidirect product $V ⋊ K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$ . Let $π$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$ . We construct an invariant symbolic calculus for $π$ , under some technical hypothesis. We give some examples including the Poincaré group. ...

Interaction between cellularity of Alexandroff spaces and entropy of generalized shift maps

Fatemah Ayatollah Zadeh Shirazi, Sahar Karimzadeh Dolatabad, Sara Shamloo (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In the following text for a discrete finite nonempty set $K$ and a self-map $ϕ : X \to X$ we investigate interaction between different entropies of generalized shift $σ_{ϕ} : K^{X} \to K^{X}$ , ${(x_{α})}_{α \in X} \mapsto {(x_{ϕ (α)})}_{α \in X}$ and cellularities of some Alexandroff topologies on $X$ .

Shadowing in actions of some Abelian groups

Sergei Yu. Pilyugin, Sergei B. Tikhomirov (2003)

Fundamenta Mathematicae

Similarity:

We study shadowing properties of continuous actions of the groups $ℤ^{p}$ and $ℤ^{p} \times ℝ^{p}$ . Necessary and sufficient conditions are given under which a linear action of $ℤ^{p}$ on $ℂ^{m}$ has a Lipschitz shadowing property.

Explicit computations of all finite index bimodules for a family of II $_{1}$ factors

Stefaan Vaes (2008)

Annales scientifiques de l'École Normale Supérieure

Similarity:

We study II $_{1}$ factors $M$ and $N$ associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result : every finite index $M$ - $N$ -bimodule (in particular, every isomorphism between $M$ and $N$ ) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index $M$ - $M$ -bimodules is identified with an extended Hecke fusion algebra,...

Corrigendum to “Braided coproduct, antipode and adjoint action for $U_{q} (s l_{2})$ ” [Arch. Math. (Brno) 60(5) (2024), 365–376, DOI: 10.5817/AM2024-5-365]

(2025)

Archivum Mathematicum

Similarity:

Minimal models for $ℤ^{d}$ -actions

Bartosz Frej, Agata Kwaśnicka (2008)

Colloquium Mathematicae

Similarity:

We prove that on a metrizable, compact, zero-dimensional space every $ℤ^{d}$ -action with no periodic points is measurably isomorphic to a minimal $ℤ^{d}$ -action with the same, i.e. affinely homeomorphic, simplex of measures.

Extensions of generic measure-preserving actions

Julien Melleray (2014)

Annales de l’institut Fourier

Similarity:

We show that, whenever $Γ$ is a countable abelian group and $Δ$ is a finitely-generated subgroup of $Γ$ , a generic measure-preserving action of $Δ$ on a standard atomless probability space $(X, μ)$ extends to a free measure-preserving action of $Γ$ on $(X, μ)$ . This extends a result of Ageev, corresponding to the case when $Δ$ is infinite cyclic.

The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

Mrinal Kanti Roychowdhury, Daniel J. Rudolph (2008)

Fundamenta Mathematicae

Similarity:

Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) ${ϕ |}_{X ₀}$ is continuous in the relative topology on X₀ and $ϕ^{- 1} |_{Y ₀}$ is continuous in the relative topology on Y₀, (2) $ϕ (O r b_{T} (x)) = O r b_{S} (ϕ (x))$ for μ-a.e. x ∈ X. (X,,μ,T)...

3x+1 inverse orbit generating functions almost always have natural boundaries

Jason P. Bell, Jeffrey C. Lagarias (2015)

Acta Arithmetica

Similarity:

The 3x+k function $T_{k} (n)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map $T_{k} (\cdot)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_{k} (\cdot)$ . We consider the generating functions $f_{k, m} (z) = \sum_{n > 0, n \to m} z^{n}$ , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions $f_{k, m} (z)$ to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold...

Matrix coefficients, counting and primes for orbits of geometrically finite groups

Amir Mohammadi, Hee Oh (2015)

Journal of the European Mathematical Society

Similarity:

Let $G : = SO {(n, 1)}^{\circ}$ and $Γ (n - 1) / 2$ for $n = 2, 3$ and when $δ > n - 2$ for $n \geq 4$ , we obtain an effective archimedean counting result for a discrete orbit of $Γ$ in a homogeneous space $H ∖ G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family ${ℬ_{T} \subset H ∖ G}$ of compact subsets, there exists $η > 0$ such that $# [e] Γ \cap ℬ_{T} = ℳ (ℬ_{T}) + O (ℳ {(ℬ_{T})}^{1 - η})$ for an explicit measure $ℳ$ on $H ∖ G$ which depends on $Γ$ . We also apply the affine sieve and describe the distribution of almost primes on orbits of $Γ$ in arithmetic...

Characterization of cycle domains via Kobayashi hyperbolicity

Gregor Fels, Alan Huckleberry (2005)

Bulletin de la Société Mathématique de France

Similarity:

A real form $G$ of a complex semi-simple Lie group $G^{ℂ}$ has only finitely many orbits in any given $G^{ℂ}$ -flag manifold $Z = G^{ℂ} / Q$ . The complex geometry of these orbits is of interest, e.g., for the associated representation theory. The open orbits $D$ generally possess only the constant holomorphic functions, and the relevant associated geometric objects are certain positive-dimensional compact complex submanifolds of $D$ which, with very few well-understood exceptions, are parameterized by the Wolf cycle...

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

Consecutive square-free values of the type $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$

Ya-Fang Feng (2023)

Czechoslovak Mathematical Journal

Similarity:

We show that for any given integer $k$ there exist infinitely many consecutive square-free numbers of the type $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$ . We also establish an asymptotic formula for $1 \leq x, y, z \leq H$ such that $x^{2} + y^{2} + z^{2} + k$ , $x^{2} + y^{2} + z^{2} + k + 1$ are square-free. The method we used in this paper is due to Tolev.

On $k$ -free numbers over Beatty sequences

Wei Zhang (2023)

Czechoslovak Mathematical Journal

Similarity:

We consider $k$ -free numbers over Beatty sequences. New results are given. In particular, for a fixed irrational number $α > 1$ of finite type $τ < \infty$ and any constant $ε > 0$ , we can show that $\sum_{\binom{1 \leq n \leq x}{[α n + β] \in 𝒬_{k}}} 1 - \frac{x}{ζ (k)} ≪ x^{k / (2 k - 1) + ε} + x^{1 - 1 / (τ + 1) + ε},$ where $𝒬_{k}$ is the set of positive $k$ -free integers and the implied constant depends only on $α,$ $ε,$ $k$ and $β .$ This improves previous results. The main new ingredient of our idea is employing double exponential sums of the type $\sum_{1 \leq h \leq H} \sum_{\binom{1 \leq n \leq x}{n \in 𝒬_{k}}} e (ϑ h n) .$

n-supercyclic and strongly n-supercyclic operators in finite dimensions

Romuald Ernst (2014)

Studia Mathematica

Similarity:

We prove that on $ℝ^{N}$ , there is no n-supercyclic operator with 1 ≤ n < ⌊(N + 1)/2⌋, i.e. if $ℝ^{N}$ has an n-dimensional subspace whose orbit under $T \in (ℝ^{N})$ is dense in $ℝ^{N}$ , then n is greater than ⌊(N + 1)/2⌋. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator $T \in (ℝ^{N})$ is strongly n-supercyclic if $ℝ^{N}$ has an n-dimensional subspace whose orbit under T is dense in $ℙ ₙ (ℝ^{N})$ , the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially...

A note on the theorems of Lusternik-Schnirelmann and Borsuk-Ulam

T. E. Barros, C. Biasi (2008)

Colloquium Mathematicae

Similarity:

Let p be a prime number and X a simply connected Hausdorff space equipped with a free $ℤ_{p}$ -action generated by $f_{p} : X \to X$ . Let $α : S^{2 n - 1} \to S^{2 n - 1}$ be a homeomorphism generating a free $ℤ_{p}$ -action on the (2n-1)-sphere, whose orbit space is some lens space. We prove that, under some homotopy conditions on X, there exists an equivariant map $F : (S^{2 n - 1}, α) \to (X, f_{p})$ . As applications, we derive new versions of generalized Lusternik-Schnirelmann and Borsuk-Ulam theorems.

Relative exactness modulo a polynomial map and algebraic $(ℂ^{p}, +)$ -actions

Philippe Bonnet (2003)

Bulletin de la Société Mathématique de France

Similarity:

Let $F = (f_{1}, ..., f_{q})$ be a polynomial dominating map from $ℂ^{n}$ to $ℂ^{q}$ . We study the quotient $𝒯^{1} (F)$ of polynomial 1-forms that are exact along the generic fibres of $F$ , by 1-forms of type $d R + \sum a_{i} d f_{i}$ , where $R, a_{1}, ..., a_{q}$ are polynomials. We prove that $𝒯^{1} (F)$ is always a torsion $ℂ [t_{1}, ..., t_{q}]$ -module. Then we determine under which conditions on $F$ we have $𝒯^{1} (F) = 0$ . As an application, we study the behaviour of a class of algebraic $(ℂ^{p}, +)$ -actions on $ℂ^{n}$ , and determine in particular when these actions are trivial.

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

Similarity:

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

Topological dynamics of unordered Ramsey structures

Moritz Müller, András Pongrácz (2015)

Fundamenta Mathematicae

Similarity:

We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow $M (G_{)}$ of the automorphism group $G$ of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of $M (G_{)}$ . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and $G$ is amenable, then $M (G_{)}$ admits...

Equalizers and coactions of groups

Martin Arkowitz, Mauricio Gutierrez (2002)

Fundamenta Mathematicae

Similarity:

If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group $_{f} \subseteq G * H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism $p_{f} {= p |}_{f} :_{f} \to G$ . A right inverse (section) $G \to_{f}$ of $p_{f}$ is called a coaction on G. In this paper we study $_{f}$ and the sections of $p_{f}$ . We consider the following topics: the structure of $_{f}$ as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and...

Displaying similar documents to “A map maintaining the orbits of a given ℤ d -action”

Displaying similar documents to “A map maintaining the orbits of a given $ℤ^{d}$ -action”