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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

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