Roots, symmetries and conjugacy of pseudo-Anosov mapping classes

An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: decide if the element commutes with a finite order element and in this case compute this element. The structure theorem on which this algorithm is based provides also a new solution to the conjugacy problem

Volume entropy for surface groups via Bowen-Series like maps

We define a Bowen-Series like map for every geometric presentation of a co-compact surface group and we prove that the volume entropy of the presentation is the topological entropy of this particular (circle) map. Finally we find the minimal volume entropy among geometric presentations

Infinite sequence of fixed point free pseudo-Anosov homeomorphisms

We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of abelian differentials. The existence of such sequences show that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy-Veech induction strategy

A note on covers of fibred hyperbolic manifolds

For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, $\varphi_1$ and $\varphi_2$, and two non-equivalent covers $p_i: \tilde S \longrightarrow S$, $i=1,2$, so that the lift of $\varphi_1$ to $\tilde S$ with respect to $p_1$ coincides with that of $\varphi_2$ with respect to $p_2$. The mapping tori of the $\varphi_i$ and their lift provide examples of pairs of hyperbolic $3$-manifolds so that the first is covered by the second in two different ways.Comment: 9 pages, 2 figure

A group from a map and orbit equivalence

In two papers published in 1979, R. Bowen and R. Bowen and C. Series introduced a dynamical system from a Fuchsian group, acting on the hyperbolic plane $\mathbb{H}^2$. The dynamics is a map on $S^1$ which is, in particular, an expanding piecewise homeomorphism of the circle. In this paper we consider a reverse question: which dynamical conditions for an expanding piecewise homeomorphism of $S^1$ are sufficient for the map to be a ``Bowen-Series-type" map (see below) for some group $G$ and which groups can occur? We give a partial answer to these questions.Comment: 46 pages, 8 figure

Geometrization of Piecewise homeomorphisms of the circle

In this work we study the following realization problem: given a piecewise homeomorphism $\Phi: S^1 \rightarrow S^1$, which geometrical and dynamical conditions on $\Phi$ are sufficient to realize it as a Bowen-Series-like map associated to a surface group, seen as a discrete subgroup of ${\rm Homeo}^+ (S^1)$?Comment: 41 pages, 12 figure

Combinatorial suspension for disc-homeomorphisms

International audienceFor a punctured-disc homeomorphism given combinatorially, we give an algorithmic construction of the suspension flow in the corresponding mapping-torus. In particular one computes explicitly the embedding in the mapping-torus of any finite collection of periodic orbits for this flow. All these orbits are realized as closed braids carried by a branched surface which we construct in the algorithm. Our construction gives a combinatorial proof of the fact that the periodic orbits of such a suspension flow are carried by a same branched-surface

Earthworm management in tropical agroecosystems

Data of 145 and 69 earthworm communities from managed and natural ecosystems, respectively, of four continents and 15 tropical countries were analysed. The aim of the study was to separate the influence of phylogenetic, environmental and agricultural factors on the structure of earthworm communities in agroecosystems, and to evaluate their relative importance in the whole soil macrofauna community. Earthworms comprise 40-90% of macrofaunal biomass in most ecosystems except for annually cropped systems. Three major conclusions were drawn from the analysis of community structure (regional analysis) : (i) crops were, independently of region, characterized by a loss of native species and by the dominance of exotic endogeics ; (ii) pastures were highly heterogenous in terms of native or exotic species dominance ; (iii) native species survived better in management ecosystems of India and Africa than in Mexico-Central America. Local analysis in selected countries indicated that, as a general rule, the intensity of agricultural practices is negatively correlated with the amount of native species and the total abundance and biomass of earthworms ; the only exception was found in the conversion of savannas to pastures, in Colombian llanos. (Résumé d'auteur

Volume entropy for minimal presentations of surface groups in all ranks

Agraïments: This work has been carried out thanks to the support of the ARCHIMEDE Labex (ANR-11-LABX- 0033).We study the volume entropy of a class of presentations (including the classical ones) for all surface groups, called minimal geomètric presentations. We rediscover a formula first obtained by Cannon and Wagreich [6] with the computation in a non published manuscrit by Cannon [5]. The result is surprising: an explicit polynomial of degree n, the rank of the group, encodes the volume entropy of all classical presentations of surface groups. The approach we use is completely different. It is based on a dynamical system construction following an idea due to Bowen and Series [3] and extended to all geometric presentations in [15]. The result is an explicit formula for the volume entropy of minimal presentations for all surface groups, showing a polynomial dependence in the rank n > 2. We prove that for a surface group Gn of rank n with a classical presentation Pn the volume entropy is log(λn), where λn is the unique real root larger than one of the polynomial x n − 2(n − 1) nX−1 j=1 x j + 1