Person: Rodríguez Sanjurjo, José Manuel
Universidad Complutense de Madrid
Faculty / Institute
Álgebra, Geometría y Topología
Geometría y Topología
Now showing 1 - 10 of 65
PublicationShape morphisms and components of movable compacta(Cambridge Univ Press, 1988-05) Rodríguez Sanjurjo, José ManuelThe author treats shape properties which movable compacta and their nonmovable components inherit from their movable components. First he shows that shape morphisms of movable compacta are completely determined by their restrictions to movable components. Then he gives a necessary and sufficient condition for a shape morphism α between movable compacta X and Y to be an isomorphism. Such a condition is given in terms of the morphisms induced by α between the components of X and Y . This result improves a result of Dydak and Segal in the case of movable compacta. Finally, he shows that the shape category of a movable compactum is completely determined by the shape category of its movable components. The shape category is a numerical shape invariant introduced by Borsuk which in the case of polyhedra agrees with the Lyusternik-Shnirelʹman category PublicationA non-continuous description of the shape category of compacta(Oxford University Press, 1989-09) Rodríguez Sanjurjo, José Manuel PublicationOn the topology of the boundary of a basin of attraction(American Mathematical Society, 2007) Sánchez Gabites, Jaime Jorge; Rodríguez Sanjurjo, José ManuelSuppose phi : M x R -> M is a continuous flow on a locally compact metrizable space M and K is an ( asymptotically stable) attractor. Let D =partial derivative A( K) be the boundary of the basin of attraction of K. In the present paper it will be shown how the Conley index of D plays an important role in determining the topological nature of D and allows one to obtain information about the global dynamics of phi in M. PublicationUnstable manifold, Conley index and fixed points of flows(Elsevier, 2014-12-01) Barge, Héctor; Rodríguez Sanjurjo, José ManuelWe study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point. PublicationOn a theorem of B. J. Ball(Polish Academy of Sciences, 1985) Rodríguez Sanjurjo, José ManuelThe author generalizes some results of Ball concerning the relationship between the shape of a locally compact metrizable space with compact components and the shape of its components. The following results are proved. Let X and Y be locally compact metrizable spaces with compact components. (1) If μ:X→Y is a shape morphism, then there exists exactly one function Λ:□(X)→□(Y) satisfying the following condition: If X0∈□(X) and Y0=Λ(X0) then there is a shape morphism μ0:X0→Y0 such that S[i(Y0,Y)]⋅μ0=μ⋅S[i(X0,X)], where S[i(Y0,Y)] is the shape morphism induced by the inclusion. Moreover, Λ is continuous and for every compact set A⊂□(X) there exists exactly one shape morphism η:p−1(A)→q−1(Λ(A)) satisfying the following condition: S[i(q−1(Λ(A)),Y)]⋅η=μ⋅S[i(p−1(A),X)]. (2) Let μ:X→Y be a shape morphism such that the induced map Λ:□(X)→□(Y) is a homeomorphism. If for each component X0 of X the unique shape morphism μ0:X0→Y0=Λ(X0) satisfying S[i(Y0,Y)]⋅μ0=μ⋅S[i(X0,X)] is an isomorphism, then μ is an isomorphism. PublicationOn a question of S. Godlewski(Symposium of General Topology, 1990) Laguna, V.F.; Morón, Manuel A.; Rodríguez Sanjurjo, José ManuelS. Godlewski [Fundam. Math. 114, 1-9 (1981; Zbl 0498.54016)] proved that if a metrizable space X is a mutational retract of X0 and X is an MANR (mutational absolute neighborhood retract), then every component of X is a mutational retract of a component of X0. In the same paper he posed the problem whether this result remains true without the hypothesis X 2 MANR. In the present paper a partial answer is given to this question. A topological space is said to belong to the class U0 if a) the quotient map X ! (X) is closed (where (X) denotes the space of components) and b) ind((X)) = 0, where ind represents the small inductive dimension. Examples of classes of spaces contained in U0 include the MANR’s and the class of the locally compact metrizable spaces with compact components. The main results in this paper are: Theorem 6. Let X0 be a metrizable space and suppose that X is a mutational retract of X0. If X 2 U0 then every component of X is a mutational retract of a component of X0. - Proposition 8. Let X0 2 U0 be a metrizable space and let us suppose that X is a mutational retract of X0. Then X 2 U0. From these two results the following consequence is derived. Corollary 9. Let X0 2 U0 be a metrizable space and suppose that X is a mutational retract of X0. Then every component of X is a mutational retract of a component of X0. This corollary solves Godlewski’s problem provided the ambient space X0 belongs to the class U0. PublicationShape of global attractors in topological spaces(Pergamon-Elsevier Science, 2005-02) Giraldo, A.; Morón, Manuel A.; Romero Ruiz del Portal, Francisco; Rodríguez Sanjurjo, José ManuelIn this paper, we apply the notion and properties of compactly generated shape to study attractors in topological spaces. PublicationFinite approximations to Cech homology(Elsevier Science B.V. (North-Holland), 2001-09-07) Giraldo, A.; Morón, Manuel A.; Romero Ruiz del Portal, Francisco; Rodríguez Sanjurjo, José ManuelWe show in this paper how to represent intrinsically Cech homology of compacta, in terms of inverse limits of discrete approximations. We establish some relations between inverse limits and non-continuous homotopies and, as a consequence, we get a strong form of the classical continuity property of Cech homology. PublicationDensity and finiteness. A discrete approach to shape(Elsevier Science, 1997-03) Giraldo, A.; Rodríguez Sanjurjo, José ManuelWe show in this paper that the category of shape can be modelled in discrete terms using maps defined in dense subsets of compacta. This approach to shape provides in addition a characterization of the shape of compacta which does not require external elements and uses only continuous single-valued functions, in contrast with the existing internal characterizations of shape. As an application, we prove a connection between the notion of shape image, due to Lisica, and the basic notion of omega limit of a dynamical system. PublicationStability of the fixed-point property and universal maps(American Mathematical Society, 1989-01) Rodríguez Sanjurjo, José ManuelIn this interesting paper, the author gives a stability condition for the fixed point property in terms of K. Borsuk's fundamental metric on a hyperspace of a compact metric space. This condition is equivalent to that originally given by V. L. Klee [Colloq. Math. 8 (1961), 43–46] but it reflects richer properties. By replacing exact conditions with their proximate analogues, the author introduces a notion of proximately universal maps and studies many of their properties. In particular, he investigates their preservation under composition with weakly refinable and refinable maps to get improvements of results of E. E. Grace [Proc. Amer. Math. Soc. 98 (1986), no. 2, 329–335] and C. W. Ho [Fund. Math. 111 (1981), no. 2, 169–177].