A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.Comment: Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue 1-2, pp 405-42

Four layer bandage compared with short stretch bandage for venous leg ulcers: systematic review and meta-analysis of randomised controlled trials with data from individual patients

<p><b>Objective:</b> To compare the effectiveness of two types of compression treatment (four layer bandage and short stretch bandage) in people with venous leg ulceration.</p> <p><b>Design:</b> Systematic review and meta-analysis of patient level data.</p> <p><b>Data:</b> sources Electronic databases (the Cochrane Central Register of Controlled Trials, the Cochrane Wounds Group Specialised Register, Medline, Embase, CINAHL, and National Research Register) and reference lists of retrieved articles searched to identify relevant trials and primary investigators. Primary investigators of eligible trials were invited to contribute raw data for re-analysis.</p> <p><b>Review:</b> methods Randomised controlled trials of four layer bandage compared with short stretch bandage in people with venous leg ulceration were eligible for inclusion. The primary outcome for the meta-analysis was time to healing. Cox proportional hazards models were run to compare the methods in terms of time to healing with adjustment for independent predictors of healing. Secondary outcomes included incidence and number of adverse events per patient.</p> <p><b>Results:</b> Seven eligible trials were identified (887 patients), and patient level data were retrieved for five (797 patients, 90% of known randomised patients). The four layer bandage was associated with significantly shorter time to healing: hazard ratio (95% confidence interval) from multifactorial model based on five trials was 1.31 (1.09 to 1.58), P=0.005. Larger ulcer area at baseline, more chronic ulceration, and previous ulceration were all independent predictors of delayed healing. Data from two trials showed no evidence of a difference in adverse event profiles between the two bandage types.</p> <p><b>Conclusions:</b> Venous leg ulcers in patients treated with four layer bandages heal faster, on average, than those of people treated with the short stretch bandage. Benefits were consistent across patients with differing prognostic profiles.</p&gt

Strictly Toral Dynamics

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus $\mathbb{T}^2$ which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points $ine(f)$ is shown to be a disjoint union of periodic topological disks ("elliptic islands"), while the set of essential points $ess(f)$ is an essential continuum, with typically rich dynamics (the "chaotic region"). This generalizes and improves a similar description by J\"ager. The key result is boundedness of these "elliptic islands", which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in $ess(f)$ is as rich as in $\mathbb{T}^2$ from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.Comment: Incorporates suggestions and corrections by the referees. To appear in Inv. Mat

Global surfaces of section in the planar restricted 3-body problem

The restricted planar three-body problem has a rich history, yet many unanswered questions still remain. In the present paper we prove the existence of a global surface of section near the smaller body in a new range of energies and mass ratios for which the Hill's region still has three connected components. The approach relies on recent global methods in symplectic geometry and contrasts sharply with the perturbative methods used until now.Comment: 11 pages, 1 figur

Cuntz-Pimsner C*-algebras associated with subshifts

By using C*-correspondences and Cuntz-Pimsner algebras, we associate to every subshift (also called a shift space) $X$ a C*-algebra $O_X$, which is a generalization of the Cuntz-Krieger algebras. We show that $O_X$ is the universal C*-algebra generated by partial isometries satisfying relations given by $X$. We also show that $O_X$ is a one-sided conjugacy invariant of $X$.Comment: 28 pages. This is a slightly updated version of a preprint from 2004. Submitted for publication. In version 2 the Introduction has been changed, two remarks (Remark 7.6 and 7.7) have been added and the list of references has been update