New Rotation Sets in a Family of Torus Homeomorphisms

We construct a family $\{\Phi_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $\rho(\Phi_t)$ can be described explicitly. We analyze the bifurcations and typical behavior of rotation sets in the family, providing insight into the general questions of toral rotation set bifurcations and prevalence. We show that there is a full measure subset of $[0,1]$, consisting of infinitely many mutually disjoint non-trivial closed intervals, on each of which the rotation set mode locks to a constant polygon with rational vertices; that the generic rotation set in the Hausdorff topology has infinitely many extreme points, accumulating on a single totally irrational extreme point at which there is a unique supporting line; and that, although $\rho(t)$ varies continuously with $t$, the set of extreme points of $\rho(t)$ does not. The family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set, or by any directional ergodic measure.Comment: Author's accepted version. The final publication is available at Springer via http://dx.doi.org/10.1007/s00222-015-0628-

Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences

A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees that the Euclidean structure on the polygons induces a unique conformal structure on the quotient surface, making it into a closed Riemann surface. In this case, a modulus of continuity for uniformizing coordinates is found which depends only on the geometry of the polygons and on the identifications. An application is presented in which a uniform modulus of continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it possible to prove that they converge to a Teichm\"uller mapping on the Riemann sphere.Comment: 75 pages, 18 figure

Unimodal generalized pseudo-Anosov maps

An infinite family of generalized pseudo-Anosov homeomorphisms of the sphere S is constructed, and their invariant foliations and singular orbits are described explicitly by means of generalized train tracks. The complex strucure induced by the invariant foliations is described, and is shown to make S into a complex sphere. The generalized pseudo-Anosovs thus become quasiconformal automorphisms of the Riemann sphere, providing a complexification of the unimodal family which differs from that of the Fatou/Julia theory.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper31.abs.htm

Curitiba: metrópole modelo ou urbe segregada? A questão habitacional e a apartação social em uma metrópole no Sul do Brasil

Referência no planejamento urbano brasileiro, Curitiba também se destaca no Brasil com bons índices econômicos e de qualidade de vida. Todavia, os diferenciais urbanos das regiões centrais da cidade contrastam com uma periferia marcada pelo baixo rendimento e informalidade habitacional. Regiões apartadas da cidade considerada “modelo” abrigam centenas de assentamentos precários e quase a totalidade da habitação de interesse social. A segregação socioespacial em Curitiba se revela especialmente na distância física e social entre as regiões que concentra alta renda àquelas de menores rendimentos. Historicamente, desde as primeiras políticas habitacionais da cidade, a população pobre foi deslocada para conjuntos habitacionais populares às margens da urbe. Mesmo após discussões e avanços legais relacionados à questão urbana e habitacional brasileira e o lançamento de um ousado programa federal habitacional em 2009, verifica-se que a prática de localizar os mais pobres nas piores e mais distantes áreas permanece, reforçando a segregação urbana.Reference in the Brazilian urban planning , Curitiba also stands out with good economic rates and quality of life . However, urban differentials of the central regions of the city contrast with a periphery marked by informality and low-income housing. Regions set apart the city considered "model " concentrate hundreds of slums and almost all of social housing. The socio-spatial segregation in Curitiba is revealed especially in the physical and social distance between regions that concentrates high income to those with lower incomes. Historically, since the first housing policies of the city, the poor population was displaced to reside the borders of the metropolis. Even after discussions and legal developments related to Brazilian urban and housing issues and launching a federal housing program in 2009, it is verified that the practice of locating the poorest in the worst and most remote areas remains, reinforcing the urban segregation.Peer Reviewe