On admissible groups of diffeomorphisms

Rybicki, Tomasz

Displaying similar documents to “On admissible groups of diffeomorphisms”

Commutators of diffeomorphisms of a manifold with boundary

Tomasz Rybicki (1998)

Annales Polonici Mathematici

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A well known theorem of Herman-Thurston states that the identity component of the group of diffeomorphisms of a boundaryless manifold is perfect and simple. We generalize this result to manifolds with boundary. Remarks on $C^{r}$ -diffeomorphisms are included.

Hyperplanes in matroids and the axiom of choice

Marianne Morillon (2022)

Commentationes Mathematicae Universitatis Carolinae

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We show that in set theory without the axiom of choice ZF, the statement sH: “Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it” implies AC $^{fin}$ , the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC $^{fin}$ in terms of “graphic” matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?

The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brelot

Peter Loeb, Bertram Walsh (1965)

Annales de l'institut Fourier

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Dans l’axiomatique des fonctions harmoniques de Brelot, où l’axiome 3 (de convergence) peut être appelé principe de Harnack, on démontre ici pour les fonctions harmoniques $> 0$ dans un domaine $ω$ valant 1 en $x_{0} \in ω$ , la propriété d’égale continuité en $x_{0}$ qui peut se traduire par des “inégalités de Harnack”. Cela avait été établi par Mokobodzki grâce à l’hypothèse d’une base dénombrable d’ouverts, qui est évitée ici en utilisant le théorème d’Éberlein-Smulian.

The gap between I₃ and the wholeness axiom

Paul Corazza (2003)

Fundamenta Mathematicae

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∃κI₃(κ) is the assertion that there is an elementary embedding $i : V_{λ} \to V_{λ}$ with critical point below λ, and with λ a limit. The Wholeness Axiom, or WA, asserts that there is a nontrivial elementary embedding j: V → V; WA is formulated in the language ∈,j and has as axioms an Elementarity schema, which asserts that j is elementary; a Critical Point axiom, which asserts that there is a least ordinal moved by j; and includes every instance of the Separation schema for j-formulas. Because no instance...

Correspondence between diffeomorphism groups and singular foliations

Tomasz Rybicki (2012)

Annales Polonici Mathematici

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It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation $ℱ_{G}$ . A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing $C^{r}$ diffeomorphism group G is simple iff the foliation $ℱ_{[G, G]}$ defined by [G,G] admits no proper...

Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

Carlos Matheus, Carlos G. Moreira, Enrique R. Pujals (2013)

Annales scientifiques de l'École Normale Supérieure

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We consider a family of planar systems introduced in 1991 by Benedicks and Carleson as a toy model for the dynamics of the so-called Hénon maps. We show that Smale’s Axiom A property is $C^{1}$ -dense among the systems in this family, despite the existence of $C^{2}$ -open subsets (closely related to the so-called Newhouse phenomena) where Smale’s Axiom A is violated. In particular, this provides some evidence towards Smale’s conjecture that Axiom A is a $C^{1}$ -dense property among surface diffeomorphisms. The...

Axioms weaker then $R_{0}$

J. Guia (1984)

Matematički Vesnik

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On the Leibniz-Mycielski axiom in set theory

Ali Enayat (2004)

Fundamenta Mathematicae

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Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set-theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that $(V_{α}, \in)$ satisfies φ(x) ∧¬ φ(y). We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows: 1. In the presence of ZF, the following are equivalent: (a)...

Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marco Marchi (2014)

Annales de l’institut Fourier

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In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type $☆$ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $☆$ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano...

Extending regular foliations

J. W. Smith (1969)

Annales de l'institut Fourier

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A $p$ -dimensional foliation $F$ on a differentiable manifold $M$ is said to extend provided there exists a $(p + 1)$ -dimensional foliation $F^{'}$ on $M$ with $F \subset F^{'}$ . Our main result asserts that if $M$ and $F$ extends over relatively compact subsets of $M$ .

Non-degenerescence of some spectral sequences

K. S. Sarkaria (1984)

Annales de l'institut Fourier

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Each Lie algebra $ℱ$ of vector fields (e.g. those which are tangent to a foliation) of a smooth manifold $M$ définies, in a natural way, a spectral sequence $E_{k} (ℱ)$ which converges to the de Rham cohomology of $M$ in a finite number of steps. We prove e.g. that for all $k \geq 0$ there exists a foliated compact manifold with $E_{k} (ℱ)$ infinite dimensional.