Discrete dynamics on noncommutative CW complexes

[EN] The concept of discrete multivalued dynamical systems for noncommutative CW complexes is developed. Stable and unstable manifolds are introduced and their role in geometric and topological configurations of noncommutative CW complexes is studied. Our technique is illustrated by an example on the noncommutative CW complex decomposition of the algebra of continuous functions on two dimensional torus.Milani, V.; Mansourbeigi, S. (2013). Discrete dynamics on noncommutative CW complexes. Applied General Topology. 14(2):179-193. doi:10.4995/agt.2013.1671.SWORD179193142Allili, M., Corriveau, D., Derivière, S., Kaczynski, T., & Trahan, A. (2007). Discrete Dynamical System Framework for Construction of Connections between Critical Regions in Lattice Height Data. Journal of Mathematical Imaging and Vision, 28(2), 99-111. doi:10.1007/s10851-007-0010-0A. Connes, Noncommutative Geometry (Academic Press, San Diego1994).S. Eilers, T.A. Loring, G.K. Pedersen, Stability of Anticommutation Relations: an application to NCCW complexes, J. Reine Angew Math. 99 (1998).Kaczynski, T., & Mrozek, M. (1995). Conley index for discrete multi-valued dynamical systems. Topology and its Applications, 65(1), 83-96. doi:10.1016/0166-8641(94)00088-kJ. Milnor, Morse Theory, Annals of Math. Studies, (Princeton Univ. Press, 1963).V. Milani, A. A. Rezaei, S. M. H. Mansourbeigi, Morse Theory for C*-Algebras: A geometric Interpretation of some Noncommutative Manifolds, Applied General Topology 12 (2011) 175-185.G.K. Pedersen, Pull Back and Pushout Constructions in C*-Algebras, J. Funct. Analysis 167 (1999).J. H. C. Whitehead, Combinatorial Homotopy, I. Bulletin of the American Society 55 (1949) 1133-1145

Stability of Reeb graphs under function perturbations: the case of closed curves

Reeb graphs provide a method for studying the shape of a manifold by encoding the evolution and arrangement of level sets of a simple Morse function defined on the manifold. Since their introduction in computer graphics they have been gaining popularity as an effective tool for shape analysis and matching. In this context one question deserving attention is whether Reeb graphs are robust against function perturbations. Focusing on 1-dimensional manifolds, we define an editing distance between Reeb graphs of curves, in terms of the cost necessary to transform one graph into another. Our main result is that changes in Morse functions induce smaller changes in the editing distance between Reeb graphs of curves, implying stability of Reeb graphs under function perturbations.Comment: 23 pages, 12 figure