Publication: E-Connectedness, Finite Approximations, Shape Theory and
Coarse Graining in Hyperspaces
Loading...
Full text at PDC
Publication Date
2008
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Exportar
Abstract
We use upper semifinite hyperspaces of compacta to describe "-connectedness and to compute homology from finite approximations. We find another connection between "-
connectedness and the so called Shape Theory. We construct a geodesically complete R-tree, by means of "-components at different resolutions, whose behavior at infinite captures the topological structure of the space of components of a given compact metric space. We also construct inverse sequences of finite spaces using internal finite approximations of compact metric spaces.
These sequences can be converted into inverse sequences of polyhedra and simplicial maps by means of what we call the Alexandroff-McCord correspondence. This correspondence allows us to relate upper semifinite hyperspaces of finite approximation with the Vietoris-Rips complexes
of such approximations at different resolutions. Two motivating examples are included in the introduction. We propose this procedure as a different mathematical foundation for problems on data analysis. This process is intrinsically related to the methodology of shape theory. Finally this paper reinforces Robins’s idea of using methods from shape theory to compute homology from finite approximations.
Description
UCM subjects
Unesco subjects
Keywords
Citation
G. Cantor. Über unendliche Lineare Punktmannifaltigkeiten. Math. Ann. 23 (1884) 453-458.
V. Robins, J.D. Meiss, E. Bradley. Computing connectedness: An exercise in computational topology. Nonlinearity. 11(4) (1998) 913-922.
V. Robins, J.D. Meiss, E. Bradley. Computing connectedness: disconnectedness and discreteness. Physica D. 139 (2000) 276-300.
V. Robins. Computational Topology at Multiple Resolutions. PhD Thesis. June 2000.
http://wwwrsphysse.anu.edu.au/ vbr110/thesis.php
V. Robins. Towards computing homology from finite approximations. Topology Proceedings, 24 (1999) 503-532.
R. Ghrist. Barcodes: The persistent homology of data. Bull. Amer. Math. Soc. (new series) 45-1 (2008) 61-75.
A. Giraldo, M.A. Morón, F.R. Ruiz del Portal, J.M.R. Sanjurjo . Finite Approximation to Cech Homology. Journal of Pure and Applied Algebra 163 (2001) 81-92.
D. Snyder. Fundamental properties of E-connectedness. Physica D. 173 (2002) 131-136.
M. Alonso-Morón, A. González Gómez. Homotopical properties of upper semifinite hyperspaces of compacta. Top. Appl. 155 (2008) 972-981.
M. Alonso-Morón, A. González Gómez. Upper semifinite hyperspaces as unifying tools in normal Hausdorff topology. Top. Appl. 154 (2007) no 10. 2142-2153.
S. Mardesic Shapes for topological spaces. Gen. Top. Appl. 3 (1973) 265-282.
S. Mardesic, J. Segal. Shape Theory. Noth-Holland . Amsterdam (1982).
J. Dydak, J. Segal. Shape Theory: An introduction. Lecture Notes in Math. 688. Springer-Verlag. Berlin (1978).
B. Hughes. Trees and ultrametric spaces: a categorical equivalence. Adv. Math. 189 (2004) no1. 265-282.
P.S. Alexandroff. Diskrete R¨aume. Matematiceski Sbornik. 2 (1937) 501-518.
M.C. McCord. Singular homology groups and homotopy groups of finite topological spaces . Duke Math. Journal 33 (1966) 465-474.
R.E. Stong. Finite topological spaces . Trans. Amer. Math. Soc. 123 (1966) 325-340.
J.P. May. Finite spaces and simplicial complexes.
http://www.math.uchicago.edu/ may/MISCMaster.html
J-C. Hausmann On the Vietoris-Rips Complexes and a Cohomology Theory for Metric Spaces. Annals of Mathematical Studies 138 (1995) 175-188.
M. Alonso-Morón, A. González Gómez. The Hausdorff metric and classifications of compacta. Bull. London Math. Soc. 38 (2006) no2. 314-322.
E. Michael. Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951) 152-182.
G. Beer. Topology on closed and closed convex sets. Mathematics and its Applications. Kluwer Academic Publishers. (1993)
S. Nadler. Hyperspaces of sets. Monographs and Textbooks in Pure and Applied Mathematics. vol. 49 Maecel Dekker, Inc. New York-Basel. (1978).
K. Borsuk. Concerning homotopy properties of compacta. Fund. Math. 62 (1968) 223-254.
K. Borsuk. Theory of Shape. Monografie Matematyczne. 59 Polish Scientific Publishers, Warszawa (1975).
H.M. Hastings . Shape theory and dynamical systems. in: N.G. Markley and W. Perizzo: The structure of attractors in dynamical systems, Lecture Notes in Math. 668, Springer-Verlag, Berlin (1978) 150-160.
H.M. Hastings . A higher-dimensional Poincaré-Bendixson theorem. Glas. Mat. Ser. III 14(34) (1979) no2, 263-268.
B.A. Bogatyi, V.I. Gutsu. On the structure of attracting compacta. Differentsial’nye Uravneniya. 25 (1989) 907-909.
B.M. Garay . Strong cellularity and global asymptotic stability. Fund. Math. 138 (1991) 147-154.
C. Tezer . Shift equivalence in homotopy. Math. Z. 210 (1992) 197-201.
B. Günter and J. Segal . Every attractor of a flow in a manifold has the shape of a finite polyhedron. Proc. Amer. Math. Soc. 119 (1993) 321-329.
J.M.R. Sanjurjo . Multihomotopy, Cech spaces of maps and shape groups. Proc. London Math. Soc. 69 (1994) 330-344.
J.M.R. Sanjurjo . On the structure of uniform attractors. J. Math. Anal. Appl. 192 (1995) 519-528.
A. Giraldo, J.M.R. Sanjurjo . On the global structure of invarinat regions of flows with asymptotically stable attractors. Math. Z. 232 (1999) 739-746.
L. Kapitandki, I. Rodnianski . Shape and Morse theory of attractors. Comm. Pure Appl. Math. 53 (2000) 218-242.
A. Giraldo, M.A. Mor´on, F.R. Ruiz del Portal, J.M.R. Sanjurjo . Some duality properties of non-saddle sets. Topology Appl. 113 (2001) 51-59.
J.M.R. Sanjurjo . Morse equations and unstable manifolds of isolated invariant sets. Nonlinearity.16 (2003) 1435-1448.
A. Giraldo, M.A. Morón, F.R. Ruiz del Portal, J.M.R. Sanjurjo . Shape of global attractors in topological spaces. Nonlinear Anal. 60 (2005) 837-847.
M.A. Mor´on, F.R. Ruiz del Portal, . A note about the shape of attractors of discrete semidynamical systems. Proc. Amer. Math. Soc. 134 (2006) 2165-2167.
M.A. Morón, J.J. Sanchez-Gabites, J.M.R. Sanjurjo . Topology and dynamics of unstable attractors. Fund. Math. 197 (2007) 239-252.
A. Giraldo, R. Jimenez, M.A. Morón, F.R. Ruiz del Portal, J.M.R. Sanjurjo . Pointed shape and global attractors for metrizable spaces. Topology Appl. To appear .
J. C. Robinson . Global attractors: topology and finite dimensional dynamics. J. Dyn. Differential Equations 11. (1999) 557-581.
J.T. Rogers. The shape of a cross-section of the solution funnel of an ordinary differential equation. Illinois J. Math. 21 (1977) 420-426.
J.W. Robbin, D. Salamon. Dynamical systems, shape theory and the Conley index . Ergodic Theory and Dynamical Systems 8 (1988) 375-393.
M. Mrozek. Shape index and other indices of Conley type for local maps in locally compact spaces Hausdorff spaces. Fund. Math. 145 (1) (1994) 15-37.
J. Dugundji. Topology. Allyn and Bacon, Boston. (1966).
J. M. R. Sanjurjo, An intrinsic description of Shape. Trans. Amer. Math. Soc. 329, 2 (1992) 625-636.
J. Latschev. Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold. Arch. Math. 77 (2001) 522-528.