doi: 10.3934/dcds.2021197
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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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.   Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.   Google Scholar

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