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

Topological stability from Gromov-Hausdorff viewpoint

Alexanger Arbieto and  Carlos Arnoldo Morales Rojas

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.

Keywords: Topological stability, topological GH-stability, metric space.
Mathematics Subject Classification: Primary: 54H20; Secondary: 53C23.
Citation: Alexanger Arbieto, Carlos Arnoldo Morales Rojas. Topological stability from Gromov-Hausdorff viewpoint. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc., 112 (1964), 483-507.  doi: 10.1090/S0002-9947-1964-0175081-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Z. Nitecki, On semi-stability for diffeomorphisms, Invent. Math., 14 (1971), 83-122.  doi: 10.1007/BF01405359.  Google Scholar

[8] Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The Press M.I.T., Cambridge, Mass.-London, 1971.   Google Scholar
[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.  Google Scholar

[10]

P. Petersen, Riemannian Geometry. Third Edition, Graduate Texts in Mathematics, 171. Springer, Cham, 2016. doi: 10.1007/978-3-319-26654-1.  Google Scholar

[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.  Google Scholar

[12]

P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.  doi: 10.1016/0040-9383(70)90051-0.  Google Scholar

[13]

K. Yano, Topologically stable homeomorphisms of the circle, Nagoya Math. J., 79 (1980), 145-149.  doi: 10.1017/S0027763000018997.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc., 112 (1964), 483-507.  doi: 10.1090/S0002-9947-1964-0175081-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Z. Nitecki, On semi-stability for diffeomorphisms, Invent. Math., 14 (1971), 83-122.  doi: 10.1007/BF01405359.  Google Scholar

[8] Z. Nitecki, Differentiable Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms, The Press M.I.T., Cambridge, Mass.-London, 1971.   Google Scholar
[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.  Google Scholar

[10]

P. Petersen, Riemannian Geometry. Third Edition, Graduate Texts in Mathematics, 171. Springer, Cham, 2016. doi: 10.1007/978-3-319-26654-1.  Google Scholar

[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.  Google Scholar

[12]

P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78.  doi: 10.1016/0040-9383(70)90051-0.  Google Scholar

[13]

K. Yano, Topologically stable homeomorphisms of the circle, Nagoya Math. J., 79 (1980), 145-149.  doi: 10.1017/S0027763000018997.  Google Scholar

Figure 1.  Topologically but not topologically GH-stable homeomorphism
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Figure 2.  Isometric stability in $S^1$ implies topological stability
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