A Classification of braid types for periodic orbits of diffeomorphisms of surfaces of genus one with topological entropy zero

We classify the braid types that can occur for finite unions of periodic orbits of diffeomorphisms of surfaces of genus one with zero topological entropy

Types d'orbites et dynamique minimale pour les applicationes continues de graphes

We define the type of a periodic orbit of a graph map. We consider the class of ‘train-track’ representatives, that is, those graph maps which minimize the topological entropy of the topological representatives of a given free group endomorphism. We prove that each type of periodic orbit realized by an efficient representative is also realised by any representative of the same free group endomorphism. Moreover, the number of periodic orbits of a given type is minimized by the efficient representatives

The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

International audienceWe study the pure braid groups $P_n(RP^2)$ of the real projective plane $RP^2$, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence $1 \to P_m(RP^2 \ {x_1,...,x_n} \to P_{n+m}(RP^2) \stackrel{p_{\ast}}{\to} P_n(RP^2) \to 1$, where $n\geq 2$ and $m\geq 1$, and $p_{\ast}$ is the homomorphism which corresponds geometrically to forgetting the last $m$ strings. This problem is equivalent to that of the existence of a section for the associated fibration $p: F_{n+m}(RP^2) \to F_n(RP^2)$ of configuration spaces. Van Buskirk proved in 1966 that $p$ and $p_{\ast}$ admit a section if $n=2$ and $m=1$. Our main result in this paper is to prove that there is no section if $n\geq 3$. As a corollary, it follows that $n=2$ and $m=1$ are the only values for which a section exists. As part of the proof, we derive a presentation of $P_n(RP^2)$: this appears to be the first time that such a presentation has been given in the literature

A Classification of braid types for periodic orbits of diffeomorphisms of surfaces of genus one with topological entropy zero

We classify the braid types that can occur for finite unions of periodic orbits of diffeomorphisms of surfaces of genus one with zero topological entropy

Types d'orbites et dynamique minimale pour les applicationes continues de graphes

We define the type of a periodic orbit of a graph map. We consider the class of ‘train-track’ representatives, that is, those graph maps which minimize the topological entropy of the topological representatives of a given free group endomorphism. We prove that each type of periodic orbit realized by an efficient representative is also realised by any representative of the same free group endomorphism. Moreover, the number of periodic orbits of a given type is minimized by the efficient representatives

A survey of the homotopy properties of inclusion of certain types of configuration spaces into the Cartesian product

International audienceLet $X$ be a topological space. In this survey we consider several types of configuration spaces, namely the classical (usual) configuration spaces $F_n(X)$ and $D_n(X)$, the orbit configuration spaces $F_n^G(X)$ and $F_n^G(X)/S_n$ with respect to a free action of a group $G$ on $X$, and the graph configuration spaces $F_n^{\Gamma}(X)$ and $F_n^{\Gamma}(X)/H$, where $\Gamma$ is a graph and $H$ is a suitable subgroup of the symmetric group $S_n$. The ordered configuration spaces $F_n(X)$, $F_n^G(X)$, $F_n^{\Gamma}(X)$ are all subsets of the $n$-fold Cartesian product $\prod_1^n\, X$ of $X$ with itself, and satisfy $F_n^G(X)\subset F_n(X) \subset F_n^{\Gamma}(X)\subset \prod_1^n\, X$. If $A$ denotes one of these configuration spaces, we analyse the difference between $A$ and $\prod_1^n\, X$ from a topological and homotopical point of view. The principal results known in the literature concern the usual configuration spaces. We are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusion $\iota \colon\thinspace A \longrightarrow \prod_1^n\, X$, the homotopy type of the homotopy fibre $I_{\iota}$ of the map $\iota$ via certain constructions on various spaces that depend on $X$, and the long exact sequence in homotopy of the fibration involving $I_{\iota}$ and arising from the inclusion $\iota$. In this respect, if $X$ is either a surface without boundary, in particular if $X$ is the $2$-sphere or the real projective plane, or a space whose universal covering is contractible, or an orbit space $\mathbb{S}^k/G$ of the $k$-dimensional sphere by a free action of a Lie Group $G$, we present some recent results obtained in [23,24] for the first case, and in [18] for the second and third cases. We briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest. In order to motivate various questions, for the remaining types of configuration spaces, we describe and prove a few of their basic properties. We finish the paper with a list of open questions and problems