Strong chain transitivity in hyperspaces of uniform spaces

Topological chain transitivity‎, topological strong chain transitivity, ‎topological average shadowing‎, topological ergodic shadowing‎, ‎uniform space, hyperspace

Supporting agencies:

This research was not funded

Abstract:

We apply topological definitions of strong chain transitivity and characterizetopological strong chain transitivity for the induced maps on hyperspaces ofuniform spaces. We prove that, if the induced maps on such hyperspaces exhibittopological strong chain transitivity then the base map satisfies the propertyof topological strong chain transitivity.

Show more Show less

References:

S. A. Ahmadi, Strong chain transitivity via uniformity, Qual. Theory Dyn. Syst. 23 (2024), 123. https://doi.org/10.1007/s12346-024-00983-4

S. A. Ahmadi, On the topology of the chain recurrent set of a dynamical system, Appl. Gen. Topol. 15, no. 2 (2014), 167-174. https://doi.org/10.4995/agt.2014.3050

S. A. Ahmadi, Shadowing, ergodic shadowing and uniform spaces, Filomat 31 (2017), 5117-5124. https://doi.org/10.2298/FIL1716117A

S. A. Ahmadi, A recurrent set for one-dimensional dynamical systems, Hacet. J. Math. Stat. 47, no. 1 (2018), 1-7.

S. A. Ahmadi and M. R. Molaei, Exponential limit shadowing, Ann. Polon. Math. 108, no. 1 (2013), 1-10. https://doi.org/10.4064/ap108-1-1

S. A. Ahmadi and X. Wu, A note on topological average shadowing property via uniformity, Qual. Theory Dyn. Syst. 22, no. 3 (2023), 84. https://doi.org/10.1007/s12346-023-00791-2

S. A. Ahmadi, X. Wu, and G. Chen, Topological chain and shadowing properties of dynamical systems on

uniform spaces, Topology Appl. 275 (2020), 107153. https://doi.org/10.1016/j.topol.2020.107153

S. A. Ahmadi, X. Wu, Z. Feng, X. Ma, and T. Lu, On the entropy points and shadowing in uniform spaces, Int. J. Bifurcation and Chaos 28 (2018), 1850155. https://doi.org/10.1142/S0218127418501559

E. Akin, The general topology of dynamical systems, volume 1 of Graduate Studies in Mathematics,

American Mathematical Society, Providence, RI, 1993.

E. Akin and J. Wiseman, Chain recurrence and strong chain recurrence on uniform spaces, Dynamical systems and random processes, volume 736 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2019), 1-29. https://doi.org/10.1090/conm/736/14831

A. Barzanouni, Some properties of strong chain transitive maps, Commun. Korean Math. Soc. 34, no. 3 (2019), 951-965.

C. Conley, Isolated invariant sets and the Morse index, volume 38 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, R.I., 1978. https://doi.org/10.1090/cbms/038

T. Das, K. Lee, D. Richeson, and J. Wiseman, Spectral decomposition for topologically anosov homeomorphisms on noncompact and non-metrizable spaces, Topology Appl. 160, no. 1 (2013), 149-158. https://doi.org/10.1016/j.topol.2012.10.010

R. W. Easton, Chain transitivity and the domain of influence of an invariant set, 1978. https://doi.org/10.1007/BFb0101782

L. Fernández, C. Good, M. Puljiz, and Á. Ramírez, Chain transitivity in hyperspaces, Chaos Solitons Frac. 81 (2015), 83-90. https://doi.org/10.1016/j.chaos.2015.08.026

C. Good, R. Leek, and J. Mitchell, Equicontinuity, transitivity and sensitivity: the Auslander-Yorke

dichotomy revisited, Discrete Contin. Dyn. Syst. 40, no. 4 (2020), 2441-2474. https://doi.org/10.3934/dcds.2020121

C. Good and S. Macías, What is topological about topological dynamics?, Discrete Contin. Dyn. Syst. 38 (2018), 1007-1031. https://doi.org/10.3934/dcds.2018043

Z. J. Ji, Dynamical property of hyperspace on uniform space, Demonstr. Math. 56, no. 1 (2023), 20230264. https://doi.org/10.1515/dema-2023-0264

Z. J. Ji, The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property, 20, no. 1 (2022), 333-340. https://doi.org/10.1515/math-2022-0026

Z. J. Ji, Topological structure of G-non recurrent point and dynamical properties of G-average shadowing property, J. Zhejiang Univ. Sci. 51, no. 3 (2024), 308-313.

J. L. Kelley, General topology, Springer-Verlag, New York-Berlin, 1975.

F. Pirfalak, S. A. Ahmadi, X. Wu, and N. Kouhestani, Topological average shadowing property on uniform spaces, Qual. Theory Dyn. Syst. 20(2), no. 31 (2021), 31. https://doi.org/10.1007/s12346-021-00466-w

M. Salman, X. Wu, and R. Das, Sensitivity of nonautonomous dynamical systems on uniform spaces, Int. J. Bifurcation and Chaos 31, no. 1 (2021), 2150017. https://doi.org/10.1142/S0218127421500176

P. Sharma, Induced dynamics on the hyperspaces, Appl. Gen. Topol. 17, no. 2 (2016), 93-104. https://doi.org/10.4995/agt.2016.4154

P. Sharma and A. Nagar, Topological dynamics on hyperspaces, Appl. Gen. Topol. 11, no. 1 (2010), 1-19. https://doi.org/10.3366/nor.2010.0001

D. Smith, Hyperspaces of a uniformizable space, Mathematical Proceedings of the Cambridge Philosophical Society 62, no. 1 (1966), 25-28. https://doi.org/10.1017/S0305004100039487

J. Wiseman, The generalized recurrent set and strong chain recurrence, Ergodic Theory and Dynamical Systems 38, no. 2 (2018), 788-800. https://doi.org/10.1017/etds.2016.35

J. Wiseman, Generalized recurrence and the nonwandering set for products, Topology Appl. 219 (2017), 111-121. https://doi.org/10.1016/j.topol.2017.01.010

X. Wu, S. Liang, X. Ma, T. Lu, and S. A. Ahmadi, The mean sensitivity and mean equicontinuity in uniform spaces, Int. J. Bifurcation and Chaos (2020), 2050122. https://doi.org/10.1142/S0218127420501229

K. S. Yan and F. P. Zeng, Topological stability and pseudo-orbit tracing property for homeomorphisms on uniform spaces, Acta Math. Sin. Engl. Ser. 38, no. 2 (2022), 431 - 442. https://doi.org/10.1007/s10114-021-0232-x

Show more Show less