Minimal models for $ℤ^{d}$-actions

Bartosz Frej; Agata Kwaśnicka

Displaying similar documents to “Minimal models for $ℤ^{d}$ -actions”

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

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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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 $ℝ^{4}$ .

On NIP and invariant measures

Ehud Hrushovski, Anand Pillay (2011)

Journal of the European Mathematical Society

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We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if $p = tp (b / A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd (A)$ , (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ ...

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

Jarosław Kwapisz (2010)

Fundamenta Mathematicae

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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.

Zero-set property of o-minimal indefinitely Peano differentiable functions

Andreas Fischer (2008)

Annales Polonici Mathematici

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Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let $^{\infty}$ denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits $^{\infty}$ cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a $^{\infty}$ function f:Rⁿ → R. This implies $^{\infty}$ approximation of definable continuous functions and gluing of $^{\infty}$ functions defined on closed definable sets.

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

Stefaan Vaes (2008)

Annales scientifiques de l'École Normale Supérieure

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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,...

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a new theorem on the differential inequality $u^{(m)} \leq w (u)$ . Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f (t, x)$ .

Spaces of functions on domain $Ω$ , whose $k$ -th derivatives are measures defined on $\bar{Ω}$

Jiří Souček (1972)

Časopis pro pěstování matematiky

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On inhomogeneous self-similar measures and their $L^{q}$ spectra

Przemysław Liszka (2013)

Annales Polonici Mathematici

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Let $S_{i} : ℝ^{d} \to ℝ^{d}$ for i = 1,..., N be contracting similarities, let $(p ₁, . . ., p_{N}, p)$ be a probability vector and let ν be a probability measure on $ℝ^{d}$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on $ℝ^{d}$ such that $μ = \sum_{i = 1}^{N} p_{i} μ \circ S_{i}^{- 1} + p ν$ . We give satisfactory estimates for the lower and upper bounds of the $L^{q}$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...

Wasserstein metric and subordination

Philippe Clément, Wolfgang Desch (2008)

Studia Mathematica

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Let $(X, d_{X})$ , $(Ω, d_{Ω})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_{p}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_{p} (X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f : Ω \to_{p} (X)$ , $ω \mapsto f_{ω}$ , be a Borel map such that f is a contraction from $(Ω, d_{Ω})$ into $(_{p} (X), W_{p})$ . Let ν₁,ν₂ be probability measures on Ω with $W_{p} (ν ₁, ν ₂)$ finite. On X we consider the subordinated measures $μ_{i} = \int_{Ω} f_{ω} d ν_{i} (ω)$ . Then $W_{p} (μ ₁, μ ₂) \leq W_{p} (ν ₁, ν ₂)$ . As an application we show that the solution measures $ϱ_{α} (t)$ ...

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

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Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets $_{ν} (X)$ and $_{ν} (X)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

Definable stratification satisfying the Whitney property with exponent 1

Beata Kocel-Cynk (2007)

Annales Polonici Mathematici

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We prove that for a finite collection of sets $A ₁, . . ., A_{s} \subset ℝ^{k + n}$ definable in an o-minimal structure there exists a compatible definable stratification such that for any stratum the fibers of its projection onto $ℝ^{k}$ satisfy the Whitney property with exponent 1.

On nearly radial marginals of high-dimensional probability measures

Bo'az Klartag (2010)

Journal of the European Mathematical Society

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Suppose that $μ$ is an absolutely continuous probability measure on $^{R}$ n, for large $n$ . Then $μ$ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if $n \geq {(C / ε)}^{C d}$ , then there exist $d$ -dimensional marginals of $μ$ that are $ε$ -far from being sphericallysymmetric, in an appropriate sense. Here $C > 0$ is a universal constant.

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

Adrian Ioana (2009)

Annales scientifiques de l'École Normale Supérieure

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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 ${SL}_{2} (ℤ)$ .

Erratum to “Geometric rigidity of $\times m$ invariant measures” (J. Eur. Math. Soc. 14, 1539–1563 (2012))

Michael Hochman (2013)

Journal of the European Mathematical Society

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Extensions of generic measure-preserving actions

Julien Melleray (2014)

Annales de l’institut Fourier

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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 sizes of the classes of $H^{(N)}$ -sets

Václav Vlasák (2014)

Fundamenta Mathematicae

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The class of $H^{(N)}$ -sets forms an important subclass of the class of sets of uniqueness for trigonometric series. We investigate the size of this class which is reflected by the family of measures (called polar) annihilating all sets from the class. The main aim of this paper is to answer in the negative a question stated by Lyons, whether the polars of the classes of $H^{(N)}$ -sets are the same for all N ∈ ℕ. To prove our result we also present a new description of $H^{(N)}$ -sets.

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

Philippe Bonnet (2003)

Bulletin de la Société Mathématique de France

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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.

Invariant densities for random $β$ -expansions

Karma Dajani, Martijn de Vries (2007)

Journal of the European Mathematical Society

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Let $β > 1$ be a non-integer. We consider expansions of the form $\sum_{i = 1}^{\infty} d_{i} / β^{i}$ , where the digits ${(d_{i})}_{i \geq 1}$ are generated by means of a Borel map $K_{β}$ defined on ${0, 1}^{ℕ} \times [0, ⌊ β ⌋ (β - 1)]$ . We show existence and uniqueness of a $K_{β}$ -invariant probability measure, absolutely continuous with respect to $m_{p} \otimes λ$ , where $m_{p}$ is the Bernoulli measure on ${0, 1}^{ℕ}$ with parameter $p$ ( $0 < p < 1$ ) and $λ$ is the normalized Lebesgue measure on $[0, ⌊ β ⌋ (β - 1)]$ . Furthermore, this measure is of the form $m_{p} \otimes μ_{β, p}$ , where $μ_{β, p}$ is equivalent to $λ$ . We prove that the measure of maximal entropy and $m_{p} \otimes λ$ are mutually...

A note on generalized projections in c₀

Beata Deręgowska, Barbara Lewandowska (2014)

Annales Polonici Mathematici

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Let V ⊂ Z be two subspaces of a Banach space X. We define the set of generalized projections by $_{V} (X, Z) : = {P \in (X, Z) : P |}_{V} = i d$ . Now let X = c₀ or $l_{\infty}^{m}$ , Z:= kerf for some f ∈ X* and $V : = Z \cap l ⁿ_{\infty}$ (n < m). The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215-227] and...

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

Convolution operators with anisotropically homogeneous measures on $ℝ^{2 n}$ with n-dimensional support

E. Ferreyra, T. Godoy, M. Urciuolo (2002)

Colloquium Mathematicae

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Let $α_{i}, β_{i} > 0$ , 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t • x = (t^{α ₁} x ₁, . . ., t^{α ₙ} x ₙ)$ , $t \circ x = (t^{β ₁} x ₁, . . ., t^{β ₙ} x ₙ)$ and $| | x | | = \sum_{i = 1}^{n} {| x_{i} |}^{1 / α_{i}}$ . Let φ₁,...,φₙ be real functions in $C^{\infty} (ℝ ⁿ - 0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on $ℝ^{2 n}$ given by $μ (E) = \int_{ℝ ⁿ} χ_{E} {(x, φ (x)) | | x | |}^{γ - α} d x$ , where $α = \sum_{i = 1}^{n} α_{i}$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{μ} f = μ * f$ and let $| | T_{μ} {| |}_{p, q}$ be the operator norm of $T_{μ}$ from $L^{p} (ℝ^{2 n})$ into $L^{q} (ℝ^{2 n})$ , where the $L^{p}$ spaces are taken with respect to the Lebesgue measure. The type set $E_{μ}$ is defined by $E_{μ} = (1 / p, 1 / q) : | | T_{μ} {| |}_{p, q} < \infty, 1 \leq p, q \leq \infty$ . In the case $α_{i} \neq β_{k}$ for 1 ≤ i,k ≤ n we characterize the...

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi, Daniel Smania (2012)

Annales scientifiques de l'École Normale Supérieure

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We consider $C^{2}$ families $t \mapsto f_{t}$ of $C^{4}$ unimodal maps $f_{t}$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure $μ_{t}$ of $f_{t}$ depends differentiably on $t$ , as a distribution of order $1$ . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of $μ_{t}$ for a Benedicks-Carleson map $f_{t}$ , in terms of a single smooth function and the...

Geometric rigidity of $\times m$ invariant measures

Michael Hochman (2012)

Journal of the European Mathematical Society

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Let $μ$ be a probability measure on $[0, 1]$ which is invariant and ergodic for $T_{a} (x) = a x 𝚖𝚘𝚍 1$ , and $0 < 𝚍𝚒𝚖 μ < 1$ . Let $f$ be a local diffeomorphism on some open set. We show that if $E \subseteq ℝ$ and $(f μ) {|_{E} \sim μ|}_{E}$ , then $f^{'} (x) \in \{\pm a^{r} : r \in ℚ\}$ at $μ$ -a.e. point $x \in f^{- 1} E$ . In particular, if $g$ is a piecewise-analytic map preserving $μ$ then there is an open $g$ -invariant set $U$ containing supp $μ$ such that $g |_{U}$ is piecewise-linear with slopes which are rational powers of $a$ . In a similar vein, for $μ$ as above, if $b$ is another integer and $a, b$ are not powers of a common integer, and if $ν$ is...

$L^{p}$ -improving properties of measures of positive energy dimension

Kathryn E. Hare, Maria Roginskaya (2005)

Colloquium Mathematicae

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A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{p}$ to $L^{q}$ for some q > p. Positive measures which are $L^{p}$ -improving are known to have positive Hausdorff dimension. We extend this result to complex $L^{p}$ -improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of $L^{p}$ -functions.

Shadowing in actions of some Abelian groups

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

Fundamenta Mathematicae

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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.

Renormings of $c_{0}$ and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this paper is to show that for every Banach space $(X, ∥ \cdot ∥)$ containing asymptotically isometric copy of the space $c_{0}$ there is a bounded, closed and convex set $C \subset X$ with the Chebyshev radius $r (C) = 1$ such that for every $k \geq 1$ there exists a $k$ -contractive mapping $T : C \to C$ with $∥ x - T x ∥ > 1 - 1 / k$ for any $x \in C$ .

Displaying similar documents to “Minimal models for ℤ d -actions”

Displaying similar documents to “Minimal models for $ℤ^{d}$ -actions”