# American Institute of Mathematical Sciences

August  2017, 37(7): 3531-3544. doi: 10.3934/dcds.2017151

## Topological stability from Gromov-Hausdorff viewpoint

 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil

Received  February 2017 Revised  March 2017 Published  March 2019

Fund Project: Work partially supported by CNPq from Brazil.

We combine the classical Gromov-Hausdorff metric [5] with the $C^0$ distance to obtain the $C^0$-Gromov-Hausdorff distance between maps of possibly different metric spaces. The latter is then combined with Walters's topological stability [11] to obtain the notion of topologically GH-stable homeomorphism. We prove that there are topologically stable homeomorphism which are not topologically GH-stable. Also that every topological GH-stable circle homeomorphism is topologically stable. Afterwards, we prove that every expansive homeomorphism with the pseudo-orbit tracing property of a compact metric space is topologically GH-stable. This is related to Walters's stability theorem [11]. Finally, we extend the topological GH-stability to continuous maps and prove the constant maps on compact homogeneous manifolds are topologically GH-stable.

Citation: Alexanger Arbieto, Carlos Arnoldo Morales Rojas. Topological stability from Gromov-Hausdorff viewpoint. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3531-3544. doi: 10.3934/dcds.2017151
##### References:
 [1] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, Recent advances. North-Holland Mathematical Library, 52. North-Holland Publishing Co. , Amsterdam, 1994. [2] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033. [3] R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc., 112 (1964), 483-507.  doi: 10.1090/S0002-9947-1964-0175081-6. [4] A. Edrei, On mappings which do not increase small distances, Proc. London Math. Soc., 2 (1952), 272-278.  doi: 10.1112/plms/s3-2.1.272. [5] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc. , Boston, MA, 1999. [6] R. Metzger, C.A. Morales and Ph. Thieullen, Topological stability in set-valued dynamics, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1965-1975.  doi: 10.3934/dcdsb.2017115. [7] Z. Nitecki, On semi-stability for diffeomorphisms, Invent. Math., 14 (1971), 83-122.  doi: 10.1007/BF01405359. [8] Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The Press M.I.T., Cambridge, Mass.-London, 1971. [9] B. Pepo, Fixed points for contractive mappings of third order in pseudo-quasimetric spaces, Indag. Math. (N.S.), 1 (1990), 473-481.  doi: 10.1016/0019-3577(90)90015-F. [10] P. Petersen, Riemannian Geometry. Third Edition, Graduate Texts in Mathematics, 171. Springer, Cham, 2016. doi: 10.1007/978-3-319-26654-1. [11] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, The structure of attractors in dynamical systems (Proc. Conf. , North Dakota State Univ. , Fargo, N. D. , 1977), Lecture Notes in Math. , Springer, Berlin, 668 (1978), 231-244. [12] P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.  doi: 10.1016/0040-9383(70)90051-0. [13] K. Yano, Topologically stable homeomorphisms of the circle, Nagoya Math. J., 79 (1980), 145-149.  doi: 10.1017/S0027763000018997.

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##### References:
 [1] N. Aoki and K. Hiraide, Topological Theory of Dynamical Systems, Recent advances. North-Holland Mathematical Library, 52. North-Holland Publishing Co. , Amsterdam, 1994. [2] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033. [3] R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc., 112 (1964), 483-507.  doi: 10.1090/S0002-9947-1964-0175081-6. [4] A. Edrei, On mappings which do not increase small distances, Proc. London Math. Soc., 2 (1952), 272-278.  doi: 10.1112/plms/s3-2.1.272. [5] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Progress in Mathematics, 152. Birkhäuser Boston, Inc. , Boston, MA, 1999. [6] R. Metzger, C.A. Morales and Ph. Thieullen, Topological stability in set-valued dynamics, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1965-1975.  doi: 10.3934/dcdsb.2017115. [7] Z. Nitecki, On semi-stability for diffeomorphisms, Invent. Math., 14 (1971), 83-122.  doi: 10.1007/BF01405359. [8] Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The Press M.I.T., Cambridge, Mass.-London, 1971. [9] B. Pepo, Fixed points for contractive mappings of third order in pseudo-quasimetric spaces, Indag. Math. (N.S.), 1 (1990), 473-481.  doi: 10.1016/0019-3577(90)90015-F. [10] P. Petersen, Riemannian Geometry. Third Edition, Graduate Texts in Mathematics, 171. Springer, Cham, 2016. doi: 10.1007/978-3-319-26654-1. [11] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, The structure of attractors in dynamical systems (Proc. Conf. , North Dakota State Univ. , Fargo, N. D. , 1977), Lecture Notes in Math. , Springer, Berlin, 668 (1978), 231-244. [12] P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.  doi: 10.1016/0040-9383(70)90051-0. [13] K. Yano, Topologically stable homeomorphisms of the circle, Nagoya Math. J., 79 (1980), 145-149.  doi: 10.1017/S0027763000018997.
Topologically but not topologically GH-stable homeomorphism
Isometric stability in $S^1$ implies topological stability
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