Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms

Let S be a compact connected surface and let f be an element of the group Homeo\_0(S) of homeomorphisms of S isotopic to the identity. Denote by \tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal cover. The homeomorphism f is said to be non-spreading if the sequence (d\_{n}/n) converges to 0, where d\_{n} is the diameter of \tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a nonempty boundary. We prove that, if S is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in Homeo\_{0}(S) arbitrarily close to the identity. In the case where the surface S is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in Homeo\_{0}(S) arbitrarily close to a rotation (this was already known in most cases by a theorem by B{\'e}guin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces S, an element of Homeo\_{0}(S) is distorted if and only if it is non-spreading

Discriminant of a generic projection of a minimal normal surface singularity

Let $(S,0)$ be a rational complex surface singularity with reduced fundamental cycle, also known as a {\em minimal} singularity. Using a fundamental result by M. Spivakovsky, we explain how to get a minimal resolution of the discriminant curve for a generic projection of $(S,0)$ onto (\C^2,0) directly from the resolution graph of $(S,0)$.Comment: February 2003. Submitted to CRA

Higher genus minimal surfaces in S3S^3 and stable bundles

We consider compact minimal surfaces $f\colon M\to S^3$ of genus 2 which are homotopic to an embedding. We assume that the associated holomorphic bundle is stable. We prove that these surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic connections are at the Weierstrass points of the Riemann surface of order at most 2. For the existence proof of the DPW potential we give a characterization of stable extensions $0\to S^{-1}\to V\to S\to 0$ of spin bundles $S$ by its dual $S^{-1}$ in terms of an associated element of $P H^0(M;K^2).$ We also consider the family of holomorphic structures associated to a minimal surface in $S^3.$ For surfaces of genus $g\geq2$ the holonomy of the connections is generically non-abelian and therefore the holomorphic structures are generically stable

Testing surface area with arbitrary accuracy

Recently, Kothari et al.\ gave an algorithm for testing the surface area of an arbitrary set $A \subset [0, 1]^n$. Specifically, they gave a randomized algorithm such that if $A$'s surface area is less than $S$ then the algorithm will accept with high probability, and if the algorithm accepts with high probability then there is some perturbation of $A$ with surface area at most $\kappa_n S$. Here, $\kappa_n$ is a dimension-dependent constant which is strictly larger than 1 if $n \ge 2$, and grows to $4/\pi$ as $n \to \infty$. We give an improved analysis of Kothari et al.'s algorithm. In doing so, we replace the constant $\kappa_n$ with $1 + \eta$ for $\eta > 0$ arbitrary. We also extend the algorithm to more general measures on Riemannian manifolds.Comment: 5 page

Competing contact processes in the Watts-Strogatz network

We investigate two competing contact processes on a set of Watts--Strogatz networks with the clustering coefficient tuned by rewiring. The base for network construction is one-dimensional chain of $N$ sites, where each site $i$ is directly linked to nodes labelled as $i\pm 1$ and $i\pm 2$. So initially, each node has the same degree $k_i=4$. The periodic boundary conditions are assumed as well. For each node $i$ the links to sites $i+1$ and $i+2$ are rewired to two randomly selected nodes so far not-connected to node $i$. An increase of the rewiring probability $q$ influences the nodes degree distribution and the network clusterization coefficient $\mathcal{C}$. For given values of rewiring probability $q$ the set $\mathcal{N}(q)=\{\mathcal{N}_1, \mathcal{N}_2, \cdots, \mathcal{N}_M \}$ of $M$ networks is generated. The network's nodes are decorated with spin-like variables $s_i\in\{S,D\}$. During simulation each $S$ node having a $D$-site in its neighbourhood converts this neighbour from $D$ to $S$ state. Conversely, a node in $D$ state having at least one neighbour also in state $D$-state converts all nearest-neighbours of this pair into $D$-state. The latter is realized with probability $p$. We plot the dependence of the nodes $S$ final density $n_S^T$ on initial nodes $S$ fraction $n_S^0$. Then, we construct the surface of the unstable fixed points in $(\mathcal{C}, p, n_S^0)$ space. The system evolves more often toward $n_S^T=1$ for $(\mathcal{C}, p, n_S^0)$ points situated above this surface while starting simulation with $(\mathcal{C}, p, n_S^0)$ parameters situated below this surface leads system to $n_S^T=0$. The points on this surface correspond to such value of initial fraction $n_S^*$ of $S$ nodes (for fixed values $\mathcal{C}$ and $p$) for which their final density is $n_S^T=\frac{1}{2}$.Comment: 5 pages, 5 figure