American Institute of Mathematical Sciences

Advanced
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 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.

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.

[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.

Figure 1.  Topologically but not topologically GH-stable homeomorphism
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Isometric stability in $S^1$ implies topological stability
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[2]

Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103
[3]

Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115
[4]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869
[5]

Qiuxia Liu, Peidong Liu. Topological stability of hyperbolic sets of flows under random perturbations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 117-127. doi: 10.3934/dcdsb.2010.13.117
[6]

H.T. Banks, S. Dediu, H.K. Nguyen. Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space. Mathematical Biosciences & Engineering, 2007, 4 (3) : 403-430. doi: 10.3934/mbe.2007.4.403
[7]

Chris Good, Sergio Macías. What is topological about topological dynamics?. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1007-1031. doi: 10.3934/dcds.2018043
[8]

Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics and Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017
[9]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . doi: 10.3934/dcds.2011.31.545
[10]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[11]

Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191
[12]

Yangrong Li, Shuang Yang, Guangqing Long. Continuity of random attractors on a topological space and fractional delayed FitzHugh-Nagumo equations with WZ-noise. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021303
[13]

Alexander J. Zaslavski. Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 245-260. doi: 10.3934/naco.2011.1.245
[14]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[15]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[16]

Luis Barreira, Claudia Valls. Topological conjugacies and behavior at infinity. Communications on Pure and Applied Analysis, 2014, 13 (2) : 687-701. doi: 10.3934/cpaa.2014.13.687
[17]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005
[18]

Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial and Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179
[19]

Jaume Llibre. Brief survey on the topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[20]

Christopher Oballe, Alan Cherne, Dave Boothe, Scott Kerick, Piotr J. Franaszczuk, Vasileios Maroulas. Bayesian topological signal processing. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 797-817. doi: 10.3934/dcdss.2021084

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (344)
  • HTML views (87)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy