548 research outputs found

    Expansive homeomorphisms of the plane

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    This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques involve topological and metric aspects of the plane. The use of a Lyapunov metric function which defines the same topology as the one induced by the usual metric but that, in general, is not equivalent to it is an example of such techniques. The discovery of a hypothesis about the behavior of Lyapunov functions at infinity allows us to generalize some results that are valid in the compact context. Additional local properties allow us to obtain another classification theorem.Comment: 29 pages, 22 figure

    LpL^p-generic cocycles have one-point Lyapunov spectrum

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    We show the sum of the first kk Lyapunov exponents of linear cocycles is an upper semicontinuous function in the LpL^p topologies, for any 1≤p≤∞1 \le p \le \infty and kk. This fact, together with a result from Arnold and Cong, implies that the Lyapunov exponents of the LpL^p-generic cocycle, p<∞p<\infty, are all equal.Comment: 8 pages. A gap in the previous version was correcte

    On the singular homology of one class of simply-connected cell-like spaces

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    In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like Peano continua, called {\sl Snake space}. In the sequel we introduced the functor SC(−,−)SC(-,-) defined on the category of all spaces with base points and continuous mappings. For the circle S1S^1, the space SC(S1,∗)SC(S^1, \ast) is a Snake space. In the present paper we study the higher-dimensional homology and homotopy properties of the spaces SC(Z,∗)SC(Z, \ast) for any path-connected compact spaces ZZ

    Continuous extension of a densely parameterized semigroup

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    Let S be a dense sub-semigroup of the positive real numbers, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over S can be extended to a weakly continuous semigroup over the positive real numbers. We obtain similar results for non-linear, non-expansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over the positive real numbers.Comment: 8 pages, minor modification

    Exploring complex networks via topological embedding on surfaces

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    We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive because any network can be embedded on a surface with sufficiently high genus. The local properties of the network are affected by the surface genus which, for example, produces significant changes in the degree distribution and in the clustering coefficient. The global properties of the graph are also strongly affected by the surface genus which is constraining the degree of interwoveness, changing the scaling properties from large-world-kind (small genus) to small- and ultra-small-world-kind (large genus). Two elementary moves allow the exploration of all networks embeddable on a given surface and naturally introduce a tool to develop a statistical mechanics description. Within such a framework, we study the properties of topologically-embedded graphs at high and low `temperatures' observing the formation of increasingly regular structures by cooling the system. We show that the cooling dynamics is strongly affected by the surface genus with the manifestation of a glassy-like freezing transitions occurring when the amount of topological disorder is low.Comment: 18 pages, 7 figure

    Topological regluing of rational functions

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    Regluing is a topological operation that helps to construct topological models for rational functions on the boundaries of certain hyperbolic components. It also has a holomorphic interpretation, with the flavor of infinite dimensional Thurston--Teichm\"uller theory. We will discuss a topological theory of regluing, and trace a direction in which a holomorphic theory can develop.Comment: 38 page

    Mappings of least Dirichlet energy and their Hopf differentials

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    The paper is concerned with mappings between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in the domain) of the energy-minimal mappings is established within the class Hˉ2(X,Y)\bar{\mathscr H}_2(X, Y) of strong limits of homeomorphisms in the Sobolev space W1,2(X,Y)W^{1,2}(X, Y), a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation leads to the Hopf differential hzhzˉˉdz⊗dzh_z \bar{h_{\bar{z}}} dz \otimes dz and its trajectories. For a pair of doubly connected domains, in which XX has finite conformal modulus, we establish the following principle: A mapping h∈Hˉ2(X,Y)h \in \bar{\mathscr H}_2(X, Y) is energy-minimal if and only if its Hopf-differential is analytic in XX and real along the boundary of XX. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in XX. Nevertheless, cracks are triggered only by the points in the boundary of YY where YY fails to be convex. The general law of formation of cracks reads as follows: Cracks propagate along vertical trajectories of the Hopf differential from the boundary of XX toward the interior of XX where they eventually terminate before making a crosscut.Comment: 51 pages, 4 figure

    Subgame-Perfection in Quitting Games with Perfect Information and Differential Equations

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    Symmetries of degenerate center singularities of plane vector fields

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    Let DD be a closed unit 22-disk on the plane centered at the origin OO, and FF be a smooth vector field such that OO is a unique singular point of FF and all other orbits of FF are simple closed curves wrapping once around OO. Thus topologically OO is a "center" singularity. Let also Diff(F)\mathrm{Diff}(F) be the group of all diffeomorphisms of DD which preserve orientation and orbits of FF. In arXiv:0907.0359 the author described the homotopy type of Diff(F)\mathrm{Diff}(F) under assumption that the 11-jet of FF at OO is non-degenerate. In this paper degenerate case is considered. Under additional "non-degeneracy assumptions" on FF the path components of Diff(F)\mathrm{Diff}(F) with respect to distinct weak topologies are described.Comment: 21 page, 3 figure

    Generic Uniqueness of Equilibrium in Large Crowding Games

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