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We classify locally finite invariant measures and orbit closure for the action of the mapping class group on the space of measured laminations on a surface. This classification translates to a classification of measures and orbit closures on the space of quadratic differentials invariant under the horospheric foliation.

We prove that the orbit closure of the determinant is not normal. A similar result is obtained for the orbit closure of the permanent multiplied by a power of a linear form., 28 pages

An interesting problem in the theory of dynamical systems is to determine the global structure of a flow from properties of characteristic invariant subsets of the phase space such as minimal sets or Poisson stable orbit closures. For flows on manifolds of dimension greater than two, the behaviour of the flow near such sets is far from being well understood. In this note we give a characterization of Denjoy flows on the torus, that is, suspensions of orientation preserving homeomorphisms of the unit circle onto itself with a Cantor minimal set, via conditions referring to the asymptotic behaviour of the orbits near a strictly Poisson stable orbit closure. More precisely, we prove the following.

In this paper we have introduced the concept of Topological Transformation Groups in fuzzy setting as a naturaltransition from the corresponding crisp structure and study some properties thereof. Classical results on orbits,orbit closure, invariant subsets are investigated in this setting. Finally we have constructed some new fuzzytopological transformation groups from given ones.

Let $G$ be a connected reductive linear algebraic group over $ℂ$ with an involution $θ$ . Denote by $K$ the subgroup of fixed points. In certain cases, the $K-orbits$ in the flag variety $G/B$ are indexed by the twisted identities $ι (θ ) = {θ (w^{-1})w | w∈W}$ in the Weyl group $W$. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph'' whose vertices form a subset of $ι (θ )$. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on $ι (θ )$ is rank symmetric. In the special case $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one'', needs to be examined. This generalises a result of Deodhar for type A Schubert varieties., Soit $G$ un groupe algébrique connexe réductif sur $ℂ$, équipé d'une involution $θ$ . Soit $K$ le sousgroupe de ses points fixes. Dans certains cas, les orbites des points de la variété de drapeaux $G/B$ sous l'action de $K$ sont indexées par les identités tordues, $ι (θ ) = {θ (w^{-1})w | w∈W}$, du groupe de Weyl $W$. Sous cette hypothèse, on établit un critère pour la lissité rationnelle des adhérences des orbites, qui généralise des résultats classiques de Carrell et Peterson pour les variétés de Schubert. Plus précisément, on peut déterminer si l'adhérence d'une orbite est rationnellement lisse en examinant les degrés dans un "Graphe de Bruhat" dont les sommets forment un sous-ensemble de $ι (θ )$. En outre, l'adhérence d'une orbite est partout rationnellement lisse si et seulement si l'intervalle correspondant dans l'ordre de Bruhat de $ι (θ )$ est symétrique respectivement au rang. Dans le cas particulier $K=\mathrm{Sp}_{2n}(ℂ), G=\mathrm{SL}_{2n}(ℂ)$, nous améliorons notre critère en montrant qu'il suffit d'examiner le degré d'un seul sommet, celui "du bas". Ceci généralise un résultat de Deodhar pour les variétés de Schubert de type A.

On the 7th, 8th and 9th June 2004 has been held at the Institut Henri Poincaré a conference on ''Dynamical systems and Diophantine approximation''. One of the aims of this conference was to give a survey of research tools at the interface between these two domains. Our goal was also to highlight methods and open questions. The proceedings we are presenting in this volume reflect the spirit of this congress. The reader will find surveys and articles on the convergence or divergence points between dynamical systems and Diophantine approximation. All these text are written to be accessible to a wide audience.

We consider a semi-simple algebraic group $\mathbf G$ defined over a local field of zero characteristic and we denote by $G$ the group of its $k$-rational points. For $\Gamma$ a "large" sub-semigroup of $G$ we define a closed subgroup 〈Spec$\Gamma$〉 associated with $\Gamma$, and we show that 〈Spec$\Gamma$〉 is large in a certain sense. This allows us to study the $\Gamma$-orbit closures for certain $\Gamma$-actions. The analytic structure of closed subgroups of $G$, over $\mathbb R$ or $\mathbb Q_{p}$, allows to use the Lie algebras techniques. The properties of the limit set of $\Gamma$ are developed ; they play an important role in the proofs.

The purpose of this research-expository paper is to clarify some of the similarities and differences between “orbit closure” and “deformations” . In Section 1 a common formulation for deformation and degeneration is presented. In particular, it is shown that if the orbit is not (Zariski) closed, then a point in the closure always arises from a deformation. In Section 2, the concept of versal deformation is introduced. It is shown that in the case of degeneration there exists a universal element. In Section 3 two definitions of “rigidity” are discussed. It is proved that in the case of Lie algebras over an algebraically closed field of characteristic 0, these two definitions are equivalent. The role of cohomology is also discussed in deformation and degeneration problems.

Let $N$ be a point of an orbit closure $\bar{O_M}$ in a module variety such that its orbit $O_N$ has codimension two in $\bar{O_M}$. We show that under some additional conditions the pointed variety $(\bar{O_M},N)$ is smoothly equivalent to a cone over a rational normal curve., 19 pages

The degeneration of controllable generalized state-space systems under transformations of restricted system equivalence is characterized. The characterization necessarily includes a characterization of the degeneration of regular matrix pencils under strict equivalence. >

In the new injection scenario the antiproton beam is injected onto a helical Tevatron orbit to avoid the detrimental effects of the beam-beam interaction at 150 GeV. The new scenario required changes in the tuning procedures. Antiprotons are too precious to be used for tuning, therefore the antiproton injection line has to be tuned with protons by reverse injecting them from the Tevatron into the Main Ring. Previously, the reverse injection was performed in one supercycle. One batch of uncoalesced bunches was injected into the Tevatron and ejected after 40 seconds. The orbit closure was performed in the Main Ring. In the new scheme the lambertson magnets have to be moved, separator polarities have to be switched, activities that cannot be completed in one supercycle. Therefore, the reverse injection sequence was changed. This involved the redefinition of TVBS (Tevatron Beam Synchronised Clock) event /sub 8/ as MRBS (Main Ring Beam Synchronized Clock) /sub 8/ thus making it possible to inject 6 proton batches (or coalesced bunches) and eject them one at a time on command, performing orbit closure each time in the Main Ring. >

Let TOEU be the tiling space on a p-dimensional subspace E of R d with a fixed lattice L by the generalized projection method. By using the dual of the lattice L we will construst explicitly a parameter family of the orbit closure decomposition of TOEU and characterize its dimension. As its application we obtain that the parameters of the orbit closure decomposition of TOEU correspond to the periods of TOE ? U, provided that TOEU and TOE ? U are given by the generalized projection method from an integral lattice L.

This paper is an introduction to toric varieties and toric resolutions. We begin with basic definitions and examples, and then cover standard topics in toric geometry, including fans, support functions, and ampleness criteria. The paper also explores alternate constructions of toric varieties and nonnormal toric varieties. Then we turn our attention to singularities. We will discuss blowing-up in the toric context and resolution of singularities for toric varieties.

International audience; In this paper we prove, using explicit constructions, that every non-compact manifold (except, of course, surfaces of finite genus) can be endowed with a non-singular flow without minimal subset, that is to say: a flow such that each orbit closure contains a smaller one.

In this chapter we consider a uniform action φ: T × X → X with X compact, and compare the semigroup and family viewpoints. Recall that by Lemma 1.2 any topological action of a uniform monoid on a compact space is a uniform action. The focus of our comparison is the uniform Stone-Ctech compactification of the uniform monoid T. In itself s u T combines three different phenomena. First T acts uniformly on the compact space s u T, which is the orbit closure of j u (0) in s u T. Furthermore using the maps Φ x , we see that this action is the universal compact T action ambit [see (6.18)]. Next s u T is an Ellis semigroup mapping onto the enveloping semigroup, S φ , by the homomorphism Φ# [see (6.13)]. Finally we recall that s u T can be regarded as the space of maximal open filters on T as in Theorem 5.2. This connects s u T with all of the family constructions in Chapters 3 and 4.

We classify the orbit closures in the variety [Formula: see text] of complex, three-dimensional Novikov algebras and obtain the Hasse diagrams for the closure ordering of the orbits. We provide invariants which are easy to compute and which enable us to decide whether or not one Novikov algebra degenerates to another Novikov algebra.

Two algorithms based on trajectory fitting have been used to obtain rapidly a circulating beam in LEP with strong focusing lattices. A new algorithm called "orbit closure" uses the beam positions measured at the first and the second turn. From their difference, it computes two corrector strengths per plane such that the second turn passes at the same place as the first turn at about ten BPM locations. This produces a closed orbit amplitude with an RMS value close to that of the first turn trajectory. A new 'first turn threader' has been tried. It detects places where an oscillation with an RMS amplitude larger than a specified value starts, and computes two corrector strengths to cancel it. By iterating this process, it should be possible to correct by steps the first turn trajectory so that its RMS amplitude is below a specified value. The orbit closure has been applied with success to a very low emittance lattice (135/spl deg/ in the horizontal plane) in LEP.

In the new injection scenario antiprotons are injected onto a helical orbit in the Tevatron in order to avoid the detrimental effects of the beam-beam interaction at 150 GeV. The new scenario required changes in the tuning procedure. Antiprotons are too precious to be used for tuning, therefore the antiproton injection line has to be tuned with protons by reverse injecting them from the Tevatron into the Main Pang (MR). Previously, the reverse injection was performed in one supercycle. One batch of uncoalesced bunches was injected into the Tevatron and ejected after 40 seconds. Then the orbit closure was performed in the MR. In the new scheme the lambertson magnets have to be moved and separator polarities have to be switched, activities that cannot be completed in one supercycle. Therefore, the reverse injection sequence was changed. This involved the redefinition of TVBS dock event $D8 as MRBS $D8 thus marking it possible to inject 6 proton batches and eject them one at a time on command, performing orbit closure each time in the MR.

For a given state space system ∑, which singular systems can occur as a limit ∑ of under high gain feedback? We identify linear systems with rectangular matrix pencils and view this question in terms of orbit closure under the strict equivalence action of GLp x GLq on the space of pxq matrix pencils. We determine necessary conditions for a singular system to be a limit of a given state space system under high gain feedback.

Each orbit closure of a recurrent discrete flow on a locally compact space is shown to contain at most one compact minimal subset. In particular, such an orbit closure contains a unique compact minimal subset if and only if the orbit closure is compact.

Two constant linear systems are said to be feedback equivalent if one can be transformed into the other via an element of the “feedback group”, which acts by state space feedback and by change of basis in the state and input spaces. Let C n,m be the space of n -dimensional completely reachable systems with m -dimensional input (pairs of matrices, n × n and n × m ). The action of the feedback group partitions the space C n,m into finitely many orbits (equivalence classes), and the closure of each orbit is a union of orbits. If one views orbit closure as ‘deformation’, then orbit closure may be considered in terms of perturbations or system failure. In this paper we determine: (1) a classification of the orbits, and (2) the orbits contained in the closure of a given orbit. Both of these problems have been solved previously (see [1,4,6,3]); here we present simple proofs and point out a connection between this problem and the analogous problem for nilpotent matrices.

Semiflows with global backward extensions are investigated. The continuity properties of the limit set and the orbit closure maps are presented. The notions of D-stability and eventual stability are studied. Stronger results are obtained for discrete and continuous semiflows

We establish some “number theoretical” results about a continuous functionh from the circleT into itself, which generalize Kronecker’s theorem in several ways. These results are used to characterize the almost periodic sets of the flow on the torusT 2 generated by (θ, φ) → (θ+α, φ+h(θ)), where α is irrational. The almost periodic measures are characterized in the caseh(θ)=θ.

We give an effective characterization of the lexicographically least word in the orbit closure of the Rudin–Shapiro word w having a specified prefix. In particular, the lexicographically least word in the orbit closure of the Rudin–Shapiro word is 0w. This answers a question Allouche et al. (Theoretical Computer Science 2009).

We construct the Hasse diagram for the closure ordering of the orbits of the irreducible reduced prehomogeneous vector space of quadruples of quinary alternating forms in five variables.