December  2019, 12(8): 2211-2220. doi: 10.3934/dcdss.2019142

Shadowing is generic on dendrites

1. 

Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223-0001, USA

2. 

Department of Mathematics, Baylor University, Waco, TX 76798-7328, USA

* Corresponding author: Will Brian

Received  August 2016 Revised  December 2016 Published  January 2019

Fund Project: The second author gratefully acknowledge support from the European Union through funding the H2020-MSCA-IF-2014 project ShadOmIC (SEP-210195797)

We show that shadowing is a generic property for continuous maps on dendrites.

Citation: Will Brian, Jonathan Meddaugh, Brian Raines. Shadowing is generic on dendrites. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2211-2220. doi: 10.3934/dcdss.2019142
References:
[1]

D. V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Curvature, Proceedings of the Steklov Institute of Mathematics, 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R. I., 1969. Google Scholar

[2]

N. C. Bernardes and U. B. Darji, Graph-theoretic structure of maps of the Cantor space, Advances in Mathematics, 231 (2012), 1655-1680. doi: 10.1016/j.aim.2012.05.024. Google Scholar

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R. E. Bowen, $ \omega$-limit sets for axiom A diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339. doi: 10.1016/0022-0396(75)90065-0. Google Scholar

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L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9. Google Scholar

[5]

R. M. Corless and S. Yu. Pilyugin, Approximate and real trajectories for generic dynamical systems, Journal of Mathematical Analysis and Applications, 189 (1995), 409-423. doi: 10.1006/jmaa.1995.1027. Google Scholar

[6]

P. KościelniakM. MazurP. Oprocha and P. Pilarczyk, Shadowing is generic-a continuous map case, Discrete and Continuous Dynamical Systems, 34 (2014), 3591-3609. doi: 10.3934/dcds.2014.34.3591. Google Scholar

[7]

M. Mazur, Weak shadowing for discrete dynamical systems on nonsmooth manifolds, Journal of Mathematical Analysis and Applications, 281 (2003), 657-662. doi: 10.1016/S0022-247X(03)00186-0. Google Scholar

[8]

M. Mazur and P. Oprocha, $ S$-limit shadowing is $ C^0$-dense, Journal of Mathematical Analysis and Applications, 408 (2013), 465-475. doi: 10.1016/j.jmaa.2013.06.004. Google Scholar

[9]

I. Mizera, Generic properties of one-dimensional dynamical systems, in Ergodic Theory and Related Topics III, volume 1514 of Lecture Notes in Mathematics, Springer, Berlin, (1992), 163-173. doi: 10.1007/BFb0097537. Google Scholar

[10]

K. Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proceedings of the American Mathematical Society, 110 (1990), 281-284. doi: 10.1090/S0002-9939-1990-1009998-8. Google Scholar

[11]

S. Yu. Pilyugin and O. B. Plamenevskaya, Shadowing is generic, Topology and Its Applications, 97 (1999), 253-266. doi: 10.1016/S0166-8641(98)00062-5. Google Scholar

[12]

K. Yano, Generic homeomorphisms of $ S^1$ have the pseudo-orbit tracing property, Journal of the Faculty of Science, University of Tokyo, Section IA: Mathematics, 34 (1987), 51-55. Google Scholar

show all references

References:
[1]

D. V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Curvature, Proceedings of the Steklov Institute of Mathematics, 90 (1967). Translated from the Russian by S. Feder, American Mathematical Society, Providence, R. I., 1969. Google Scholar

[2]

N. C. Bernardes and U. B. Darji, Graph-theoretic structure of maps of the Cantor space, Advances in Mathematics, 231 (2012), 1655-1680. doi: 10.1016/j.aim.2012.05.024. Google Scholar

[3]

R. E. Bowen, $ \omega$-limit sets for axiom A diffeomorphisms, Journal of Differential Equations, 18 (1975), 333-339. doi: 10.1016/0022-0396(75)90065-0. Google Scholar

[4]

L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4364-9. Google Scholar

[5]

R. M. Corless and S. Yu. Pilyugin, Approximate and real trajectories for generic dynamical systems, Journal of Mathematical Analysis and Applications, 189 (1995), 409-423. doi: 10.1006/jmaa.1995.1027. Google Scholar

[6]

P. KościelniakM. MazurP. Oprocha and P. Pilarczyk, Shadowing is generic-a continuous map case, Discrete and Continuous Dynamical Systems, 34 (2014), 3591-3609. doi: 10.3934/dcds.2014.34.3591. Google Scholar

[7]

M. Mazur, Weak shadowing for discrete dynamical systems on nonsmooth manifolds, Journal of Mathematical Analysis and Applications, 281 (2003), 657-662. doi: 10.1016/S0022-247X(03)00186-0. Google Scholar

[8]

M. Mazur and P. Oprocha, $ S$-limit shadowing is $ C^0$-dense, Journal of Mathematical Analysis and Applications, 408 (2013), 465-475. doi: 10.1016/j.jmaa.2013.06.004. Google Scholar

[9]

I. Mizera, Generic properties of one-dimensional dynamical systems, in Ergodic Theory and Related Topics III, volume 1514 of Lecture Notes in Mathematics, Springer, Berlin, (1992), 163-173. doi: 10.1007/BFb0097537. Google Scholar

[10]

K. Odani, Generic homeomorphisms have the pseudo-orbit tracing property, Proceedings of the American Mathematical Society, 110 (1990), 281-284. doi: 10.1090/S0002-9939-1990-1009998-8. Google Scholar

[11]

S. Yu. Pilyugin and O. B. Plamenevskaya, Shadowing is generic, Topology and Its Applications, 97 (1999), 253-266. doi: 10.1016/S0166-8641(98)00062-5. Google Scholar

[12]

K. Yano, Generic homeomorphisms of $ S^1$ have the pseudo-orbit tracing property, Journal of the Faculty of Science, University of Tokyo, Section IA: Mathematics, 34 (1987), 51-55. Google Scholar

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