September  2017, 37(9): 4943-4957. doi: 10.3934/dcds.2017212

On averaged tracing of periodic average pseudo orbits

1. 

Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland

2. 

National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22,70103 Ostrava, Czech Republic

3. 

School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China

4. 

Institute of Nonlinear Dynamics, Southwest Petroleum University, Chengdu, Sichuan 610500, China

* Corresponding author: Xinxing Wu

Received  June 2016 Revised  April 2017 Published  June 2017

Fund Project: Research of P. Oprocha was partly supported by the project "LQ1602 IT4Innovations excellence in science" and by AGH local grant. Research of X. Wu was supported by the National Natural Science Foundation of China (No. 11601449) and the scientific research starting project of Southwest Petroleum University (No. 2015QHZ029)

We propose a definition of average tracing of finite pseudo-orbits and show that in the case of this definition measure center has the same property as nonwandering set for the classical shadowing property. We also show that the average shadowing property trivializes in the case of mean equicontinuous systems, and that it implies distributional chaos when measure center is nondegenerate.

Citation: Piotr Oprocha, Xinxing Wu. On averaged tracing of periodic average pseudo orbits. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4943-4957. doi: 10.3934/dcds.2017212
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. Google Scholar

[2]

M. L. Blank, Metric properties of $\varepsilon $-trajectories of dynamical systems with stochastic behaviour, Ergodic Theory and Dynamical Systems, 8 (1988), 365-378. doi: 10.1017/S014338570000451X. Google Scholar

[3]

M. L. Blank, Deterministic properties of stochastically perturbed dynamical systems, Teor. Veroyatnost. i Primenen., 33 (1988), 659-671 (in Russian); English transl.: Theory Probab. Appl., 33 (1988), 612-623. doi: 10.1137/1133095. Google Scholar

[4]

R. Bowen, Topological entropy and axiom A, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. ⅩⅣ, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R. I., 23-41. Google Scholar

[5]

A. Fakhari and F. H. Gane, On shadowing: Ordinary and ergodic, J. Math. Anal. Appl., 364 (2010), 151-155. doi: 10.1016/j.jmaa.2009.11.004. Google Scholar

[6]

M. KulczyckiD. Kwietniak and P. Oprocha, On almost specification and average shadowing properties, Fund. Math., 224 (2014), 241-278. doi: 10.4064/fm224-3-4. Google Scholar

[7]

J. LiS. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems, 35 (2015), 2587-2612. doi: 10.1017/etds.2014.41. Google Scholar

[8]

P. Oprocha, Coherent lists and chaotic sets, Discrete Contin. Dynam. Systems, 31 (2011), 797-825. doi: 10.3934/dcds.2011.31.797. Google Scholar

[9]

P. Oprocha, Specification properties and dense distributional chaos, Discrete Contin. Dynam. Systems, 17 (2007), 821-833. doi: 10.3934/dcds.2007.17.821. Google Scholar

[10]

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

[11]

P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere, Proc. Amer. Math. Soc., 136 (2008), 3931-3940. doi: 10.1090/S0002-9939-08-09602-0. Google Scholar

[12]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory and Dynamical Systems, 27 (2007), 929-956. doi: 10.1017/S0143385706000824. Google Scholar

[13]

D. Richeson and J. Wiseman, Chain recurrence rates and topological entropy, Topology Appl., 156 (2008), 251-261. doi: 10.1016/j.topol.2008.07.005. Google Scholar

[14]

K. Sakai, Diffeomorphisms with the average-shadowing property on two-dimensional closed manifolds, Rocky Mountain J. Math., 30 (2000), 1129-1137. doi: 10.1216/rmjm/1021477263. Google Scholar

[15]

B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.1090/S0002-9947-1994-1227094-X. Google Scholar

[16]

D. Thompson, Irregular sets, the β-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414. doi: 10.1090/S0002-9947-2012-05540-1. Google Scholar

[17]

L. Wang, X. Wang, F. Lei and H. Liu, Asymptotic average shadowing property, almost specification property and distributional chaos Mod. Phys. Lett. B, 30 (2016), 1650001 [9 pages]. doi: 10.1142/S0217984916500019. Google Scholar

[18]

X. WuP. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (2016), 1942-1972. doi: 10.1088/0951-7715/29/7/1942. Google Scholar

[19]

X. Wu, L. Wang and J. Liang, The chain properties and average shadowing property of iterated function systems Qual. Theory Dyn. Syst. , DOI 10.1007/s12346-016-0220-1 (online publication). doi: 10.1007/s12346-016-0220-1. Google Scholar

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. Google Scholar

[2]

M. L. Blank, Metric properties of $\varepsilon $-trajectories of dynamical systems with stochastic behaviour, Ergodic Theory and Dynamical Systems, 8 (1988), 365-378. doi: 10.1017/S014338570000451X. Google Scholar

[3]

M. L. Blank, Deterministic properties of stochastically perturbed dynamical systems, Teor. Veroyatnost. i Primenen., 33 (1988), 659-671 (in Russian); English transl.: Theory Probab. Appl., 33 (1988), 612-623. doi: 10.1137/1133095. Google Scholar

[4]

R. Bowen, Topological entropy and axiom A, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. ⅩⅣ, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R. I., 23-41. Google Scholar

[5]

A. Fakhari and F. H. Gane, On shadowing: Ordinary and ergodic, J. Math. Anal. Appl., 364 (2010), 151-155. doi: 10.1016/j.jmaa.2009.11.004. Google Scholar

[6]

M. KulczyckiD. Kwietniak and P. Oprocha, On almost specification and average shadowing properties, Fund. Math., 224 (2014), 241-278. doi: 10.4064/fm224-3-4. Google Scholar

[7]

J. LiS. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems, 35 (2015), 2587-2612. doi: 10.1017/etds.2014.41. Google Scholar

[8]

P. Oprocha, Coherent lists and chaotic sets, Discrete Contin. Dynam. Systems, 31 (2011), 797-825. doi: 10.3934/dcds.2011.31.797. Google Scholar

[9]

P. Oprocha, Specification properties and dense distributional chaos, Discrete Contin. Dynam. Systems, 17 (2007), 821-833. doi: 10.3934/dcds.2007.17.821. Google Scholar

[10]

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

[11]

P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere, Proc. Amer. Math. Soc., 136 (2008), 3931-3940. doi: 10.1090/S0002-9939-08-09602-0. Google Scholar

[12]

C.-E. Pfister and W. G. Sullivan, On the topological entropy of saturated sets, Ergodic Theory and Dynamical Systems, 27 (2007), 929-956. doi: 10.1017/S0143385706000824. Google Scholar

[13]

D. Richeson and J. Wiseman, Chain recurrence rates and topological entropy, Topology Appl., 156 (2008), 251-261. doi: 10.1016/j.topol.2008.07.005. Google Scholar

[14]

K. Sakai, Diffeomorphisms with the average-shadowing property on two-dimensional closed manifolds, Rocky Mountain J. Math., 30 (2000), 1129-1137. doi: 10.1216/rmjm/1021477263. Google Scholar

[15]

B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.1090/S0002-9947-1994-1227094-X. Google Scholar

[16]

D. Thompson, Irregular sets, the β-transformation and the almost specification property, Trans. Amer. Math. Soc., 364 (2012), 5395-5414. doi: 10.1090/S0002-9947-2012-05540-1. Google Scholar

[17]

L. Wang, X. Wang, F. Lei and H. Liu, Asymptotic average shadowing property, almost specification property and distributional chaos Mod. Phys. Lett. B, 30 (2016), 1650001 [9 pages]. doi: 10.1142/S0217984916500019. Google Scholar

[18]

X. WuP. Oprocha and G. Chen, On various definitions of shadowing with average error in tracing, Nonlinearity, 29 (2016), 1942-1972. doi: 10.1088/0951-7715/29/7/1942. Google Scholar

[19]

X. Wu, L. Wang and J. Liang, The chain properties and average shadowing property of iterated function systems Qual. Theory Dyn. Syst. , DOI 10.1007/s12346-016-0220-1 (online publication). doi: 10.1007/s12346-016-0220-1. Google Scholar

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