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May  2022, 42(5): 2439-2451. doi: 10.3934/dcds.2021197

Shadowing as a structural property of the space of dynamical systems

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

Received  July 2021 Revised  November 2021 Published  May 2022 Early access  December 2021

We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate that, for this class of spaces, in order to determine whether a system has shadowing, it is sufficient to check that continuously generated pseudo-orbits can be shadowed.

Citation: Jonathan Meddaugh. Shadowing as a structural property of the space of dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2439-2451. doi: 10.3934/dcds.2021197
References:
[1]

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

[2]

R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747.  doi: 10.2307/2373370.

[3]

W. R. BrianJ. Meddaugh and B. E. Raines, Chain transitivity and variations of the shadowing property, Ergodic Theory Dynam. Systems, 35 (2015), 2044-2052.  doi: 10.1017/etds.2014.21.

[4]

W. R. BrianJ. Meddaugh and B. E. Raines, Shadowing is generic on dendrites, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2211-2220.  doi: 10.3934/dcdss.2019142.

[5]

R. M. Corless and S. Y. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl., 189 (1995), 409-423.  doi: 10.1006/jmaa.1995.1027.

[6]

S. Dolecki and S. Rolewicz, Metric characterizations of upper semicontinuity, J. Math. Anal. Appl., 69 (1979), 146-152.  doi: 10.1016/0022-247X(79)90184-7.

[7]

C. D. Horvath, Extension and selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse Math., 2 (1993), 253-269.  doi: 10.5802/afst.766.

[8]

P. KościelniakM. MazurP. Oprocha and P. Pilarczyk, Shadowing is generic–-a continuous map case, Discrete Contin. Dyn. Syst., 34 (2014), 3591-3609.  doi: 10.3934/dcds.2014.34.3591.

[9]

K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete Contin. Dyn. Syst., 13 (2005), 533-540.  doi: 10.3934/dcds.2005.13.533.

[10]

J. Meddaugh, On genericity of shadowing in one dimension, Fund. Math., 255 (2021), 1-18.  doi: 10.4064/fm710-11-2020.

[11]

J. Meddaugh and B. E. Raines, Shadowing and internal chain transitivity, Fund. Math., 222 (2013), 279-287.  doi: 10.4064/fm222-3-4.

[12]

P. Oprocha, Shadowing, thick sets and the Ramsey property, Ergodic Theory Dynam. Systems, 36 (2016), 1582-1595.  doi: 10.1017/etds.2014.130.

[13]

P. OprochaD. Dastjerdi and M. Hosseini, On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl., 404 (2013), 47-56.  doi: 10.1016/j.jmaa.2013.02.068.

[14]

D. W. Pearson, Shadowing and prediction of dynamical systems, Math. Comput. Modelling, 34 (2001), 813-820.  doi: 10.1016/S0895-7177(01)00101-7.

[15]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999.

[16]

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

[17]

C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 7 (1977), 425-437.  doi: 10.1216/RMJ-1977-7-3-425.

[18]

J. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89. 

[19]

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. 

show all references

References:
[1]

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

[2]

R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math., 92 (1970), 725-747.  doi: 10.2307/2373370.

[3]

W. R. BrianJ. Meddaugh and B. E. Raines, Chain transitivity and variations of the shadowing property, Ergodic Theory Dynam. Systems, 35 (2015), 2044-2052.  doi: 10.1017/etds.2014.21.

[4]

W. R. BrianJ. Meddaugh and B. E. Raines, Shadowing is generic on dendrites, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 2211-2220.  doi: 10.3934/dcdss.2019142.

[5]

R. M. Corless and S. Y. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl., 189 (1995), 409-423.  doi: 10.1006/jmaa.1995.1027.

[6]

S. Dolecki and S. Rolewicz, Metric characterizations of upper semicontinuity, J. Math. Anal. Appl., 69 (1979), 146-152.  doi: 10.1016/0022-247X(79)90184-7.

[7]

C. D. Horvath, Extension and selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse Math., 2 (1993), 253-269.  doi: 10.5802/afst.766.

[8]

P. KościelniakM. MazurP. Oprocha and P. Pilarczyk, Shadowing is generic–-a continuous map case, Discrete Contin. Dyn. Syst., 34 (2014), 3591-3609.  doi: 10.3934/dcds.2014.34.3591.

[9]

K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete Contin. Dyn. Syst., 13 (2005), 533-540.  doi: 10.3934/dcds.2005.13.533.

[10]

J. Meddaugh, On genericity of shadowing in one dimension, Fund. Math., 255 (2021), 1-18.  doi: 10.4064/fm710-11-2020.

[11]

J. Meddaugh and B. E. Raines, Shadowing and internal chain transitivity, Fund. Math., 222 (2013), 279-287.  doi: 10.4064/fm222-3-4.

[12]

P. Oprocha, Shadowing, thick sets and the Ramsey property, Ergodic Theory Dynam. Systems, 36 (2016), 1582-1595.  doi: 10.1017/etds.2014.130.

[13]

P. OprochaD. Dastjerdi and M. Hosseini, On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl., 404 (2013), 47-56.  doi: 10.1016/j.jmaa.2013.02.068.

[14]

D. W. Pearson, Shadowing and prediction of dynamical systems, Math. Comput. Modelling, 34 (2001), 813-820.  doi: 10.1016/S0895-7177(01)00101-7.

[15]

S. Y. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes in Mathematics, 1706. Springer-Verlag, Berlin, 1999.

[16]

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

[17]

C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math., 7 (1977), 425-437.  doi: 10.1216/RMJ-1977-7-3-425.

[18]

J. G. Sinaĭ, Markov partitions and U-diffeomorphisms, Funkcional. Anal. i Priložen, 2 (1968), 64-89. 

[19]

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. 

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