Shadowing in countable state shifts over compact alphabets
DOI:
https://doi.org/10.12775/TMNA.2025.063Keywords
Set-valued functions, shift spaces, symbolic dynamics, mixing properties, shadowingAbstract
We investigate shadowing and other dynamical properties of shift spaces generated by upper semi-continuous set-valued functions over a compact countable domain. We demonstrate that when the underlying metric on a countable alphabet is not the discrete metric, then it is possible for a shift of finite order to fail to have the shadowing property; this is in contrast to the fact that all shifts of finite type and all shifts of finite order over the product of the discrete metric have shadowing. We provide sufficient conditions for our shifts of order two to have the shadowing property. It is also shown that for these shifts of order two, the notions of total transitivity, topological weak mixing, and topological mixing are equivalent, even without the assumption of shadowing.References
E. Akin, E. Glasner and B. Weiss, Generically there is but one self homeomorphismof the Cantor set, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3613–3630.
L. Alvin, S. Greenwood and J.P. Kelly, Cantor sets as generalized inverse limits,Fund. Math. 266(2024), no. 1, 1–24.
L. Alvin and J.P. Kelly, Topological entropy of Markov set-valued functions, ErgodicTheory Dynam. Systems 41 (2021), no. 2, 321–337.
I. Banič, G. Erceg, S. Greenwood and J. Kennedy, Transitive points in CR-dynamicalsystems, Topology Appl. 326 (2023), paper no. 108407, 31.
I. Banič, G. Erceg, R. Gril Rogina and J. Kennedy, Minimal dynamical systems withclosed relations, Glas. Mat. Ser. III 59 (2024), no. 79 (2), 479–505.
I. Banič, G. Erceg and J. Kennedy, Closed relations with non-zero entropy that generate no periodic points, Discrete Contin. Dyn. Syst. 42 (2022), no. 10, 5137–5166.
I. Banič, R. Gril Rogina, J. Kennedy and V. Nall, Sufficient conditions for non-zero entropy of closed relations, Ergodic Theory Dynam. Systems 44 (2024), no. 11, 3091–3119.
N.C. Bernardes, Jr. and U.B. Darji, Graph theoretic structure of maps of the Cantor space, Adv. Math. 231 (2012), no. 3–4, 1655–1680.
J. Camargo and S. Macı́as, Dynamics with set-valued functions and coselections, Qual. Theory Dyn. Syst. 21 (2022), no. 2, paper no. 25, 43.
F. Capulı́n, Y. Garcı́a, D. Maya and F. Orozco-Zitli, Dynamical properties of upper
U.B. Darji, D. Gonçalves and M. Sobottka, Shadowing, finite order shifts and ultrametric spaces, Adv. Math. 385 (2021), paper no. 107760, 34.
J.-M. Gambaudo and M. Martens, Algebraic topology for minimal Cantor sets, Ann. H. Poincaré 7 (2006), no. 3, 423–446.
C. Good and J. Meddaugh, Shifts of finite type as fundamental objects in the theory of shadowing, Invent. Math. 220 (2020), no. 3, 715–736.
W.T. Ingram and W.S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math. 32 (2006), no. 1, 119–130.
J.P. Kelly and T. Tennant, Topological entropy of set-valued functions, Houston J. Math. 43 (2017), no. 1, 263–282.
B.P. Kitchens, Symbolic Dynamics, One-Sided, Two-Sided and Countable State Markov Shifts, Universitext, Springer-Verlag, Berlin, 1998.
D. Kwietniak and P. Oprocha, A note on the average shadowing property for expansive maps, Topology Appl. 159 (2012), no. 1, 19–27.
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Mathematical Library, Cambridge University Press, Cambridge, second edition, 2021.
W.S. Mahavier, Inverse limits with subsets of [0, 1] × [0, 1], Topology Appl. 141 (2004), no. 1–3, 225–231.
K. Petersen, Ergodic Theory, vol. 2 Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983.
S.Y. Pilyugin and J. Rieger, Shadowing and inverse shadowing in set-valued dynamical systems. Contractive case, Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 139–149.
S.Y. Pilyugin and J. Rieger, Shadowing and inverse shadowing in set-valued dynamical systems. Hyperbolic case, Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 151–164.
T. Shimomura, Special homeomorphisms and approximation for Cantor systems, Topology Appl. 161 (2014), 178–195.
K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc. 190 (1974), 285–299.
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 Mathematics, vol. 668, Springer, Berlin, New York, 1978 pp. 231–244.
Y.Zhang and Y.J. Zhu, Topological stability and entropy for certain set-valued maps,Acta Math. Sin. (Engl. Ser.) 40 (2024), no. 4, 962–984.
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