Abstract:
It is well known from the homotopy theory of surfaces that an ambient isotopy does not change the homotopy type of a closed curve. Using the language of dynamical systems, this means that an arc in the space of diffeomorphisms that joins two isotopic diffeomorphisms with invariant closed curves in distinct homotopy classes must go through bifurcations. A scenario is described which changes the homotopy type of the closure of the invariant manifold of a saddle point of a polar diffeomorphism of a 2-torus to any
prescribed homotopically nontrivial type. The arc constructed in the process is stable and does not change the topological conjugacy class of the original diffeomorphism. The ideas that are proposed here for constructing such an arc for a 2-torus can naturally be generalized to surfaces of greater genus.
Bibliography: 32 titles.
The investigation of the dynamics of diffeomorphisms in the class under consideration was supported by the Russian Science Foundation under grant no. 17-11-01041, the local changes of the dynamics by arcs without bifurcations were performed with the support of the Theoretical Physics and Mathematics Advancement Foundation “BASIS”, the construction of the arc was carried out at the International Laboratory of Dynamical Systems and Applications of the HSE University with the support of the Government of the Russian Federation (agreement no. 075-15-2019-1931).
Citation:
E. V. Nozdrinova, O. V. Pochinka, “Bifurcations changing the homotopy type of the closure of an invariant saddle manifold of a surface diffeomorphism”, Sb. Math., 213:3 (2022), 357–384
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\by E.~V.~Nozdrinova, O.~V.~Pochinka
\paper Bifurcations changing the homotopy type of the closure of an invariant saddle manifold of a~surface diffeomorphism
\jour Sb. Math.
\yr 2022
\vol 213
\issue 3
\pages 357--384
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This publication is cited in the following 1 articles:
A.A. Nozdrinov, E.V. Nozdrinova, O.V. Pochinka, “Stable isotopy connectivity of gradient-like diffeomorphisms of 2-torus”, Journal of Geometry and Physics, 2024, 105352
Citation in format AMSBIB
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