Expansivity and unique shadowing

Good, Chris; Macías, Sergio; Meddaugh, Jonathan; Mitchell, Joel; Thomas, Joe

doi:10.1090/proc/15204

Expansivity and unique shadowing
HTML articles powered by AMS MathViewer

by Chris Good, Sergio Macías, Jonathan Meddaugh, Joel Mitchell and Joe Thomas PDF

Proc. Amer. Math. Soc. 149 (2021), 671-685 Request permission

Abstract:

Let $f\colon X\to X$ be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map $f$ is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing, and two-sided s-limit shadowing are equivalent. We show that $f$ is positively expansive and has shadowing if and only if it has unique shadowing (i.e., each pseudo-orbit is shadowed by a unique point), extending a result implicit in Walter’s proof that positively expansive maps with shadowing are topologically stable. We use the aforementioned result on two-sided shadowing to find an equivalent characterisation of shadowing and expansivity and extend these results to the notion of $n$-expansivity due to Morales.

References

Andrew D. Barwell, Gareth Davies, and Chris Good, On the $\omega$-limit sets of tent maps, Fund. Math. 217 (2012), no. 1, 35–54. MR 2914921, DOI 10.4064/fm217-1-4
Andrew D. Barwell, Chris Good, and Piotr Oprocha, Shadowing and expansivity in subspaces, Fund. Math. 219 (2012), no. 3, 223–243. MR 3001240, DOI 10.4064/fm219-3-2
Andrew D. Barwell, Chris Good, Piotr Oprocha, and Brian E. Raines, Characterizations of $\omega$-limit sets in topologically hyperbolic systems, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 1819–1833. MR 3002729, DOI 10.3934/dcds.2013.33.1819
Andrew D. Barwell, Jonathan Meddaugh, and Brian E. Raines, Shadowing and $\omega$-limit sets of circular Julia sets, Ergodic Theory Dynam. Systems 35 (2015), no. 4, 1045–1055. MR 3345163, DOI 10.1017/etds.2013.94
Andrew D. Barwell and Brian E. Raines, The $\omega$-limit sets of quadratic Julia sets, Ergodic Theory Dynam. Systems 35 (2015), no. 2, 337–358. MR 3316915, DOI 10.1017/etds.2013.54
Rufus Bowen, Markov partitions for Axiom $\textrm {A}$ diffeomorphisms, Amer. J. Math. 92 (1970), 725–747. MR 277003, DOI 10.2307/2373370
Rufus Bowen, $\omega$-limit sets for axiom $\textrm {A}$ diffeomorphisms, J. Differential Equations 18 (1975), no. 2, 333–339. MR 413181, DOI 10.1016/0022-0396(75)90065-0
W. R. Brian, Ramsey shadowing and minimal points, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2697–2703. MR 3477087, DOI 10.1090/proc/12884
Will Brian and Piotr Oprocha, Ultrafilters and Ramsey-type shadowing phenomena in topological dynamics, Israel J. Math. 227 (2018), no. 1, 423–453. MR 3846330, DOI 10.1007/s11856-018-1739-4
William R. Brian, Jonathan Meddaugh, and Brian E. Raines, Chain transitivity and variations of the shadowing property, Ergodic Theory Dynam. Systems 35 (2015), no. 7, 2044–2052. MR 3394106, DOI 10.1017/etds.2014.21
B. Carvalho and W. Cordeiro, $n$-expansive homeomorphisms with the shadowing property, J. Differential Equations 261 (2016), no. 6, 3734–3755. MR 3527644, DOI 10.1016/j.jde.2016.06.003
Bernardo Carvalho, Hyperbolicity, transitivity and the two-sided limit shadowing property, Proc. Amer. Math. Soc. 143 (2015), no. 2, 657–666. MR 3283652, DOI 10.1090/S0002-9939-2014-12250-7
Bernardo Carvalho, Product Anosov diffeomorphisms and the two-sided limit shadowing property, Proc. Amer. Math. Soc. 146 (2018), no. 3, 1151–1164. MR 3750227, DOI 10.1090/proc/13790
Bernardo Carvalho and Welington Cordeiro, Positively $n$-expansive homeomorphisms and the L-shadowing property, J. Dynam. Differential Equations 31 (2019), no. 2, 1005–1016. MR 3951833, DOI 10.1007/s10884-018-9698-3
Bernardo Carvalho and Dominik Kwietniak, On homeomorphisms with the two-sided limit shadowing property, J. Math. Anal. Appl. 420 (2014), no. 1, 801–813. MR 3229854, DOI 10.1016/j.jmaa.2014.06.011
Robert M. Corless, Defect-controlled numerical methods and shadowing for chaotic differential equations, Phys. D 60 (1992), no. 1-4, 323–334. Experimental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991). MR 1195611, DOI 10.1016/0167-2789(92)90249-M
Robert M. Corless and S. Yu. Pilyugin, Approximate and real trajectories for generic dynamical systems, J. Math. Anal. Appl. 189 (1995), no. 2, 409–423. MR 1312053, DOI 10.1006/jmaa.1995.1027
Ethan M. Coven, Ittai Kan, and James A. Yorke, Pseudo-orbit shadowing in the family of tent maps, Trans. Amer. Math. Soc. 308 (1988), no. 1, 227–241. MR 946440, DOI 10.1090/S0002-9947-1988-0946440-2
Ethan M. Coven and Michael Keane, Every compact metric space that supports a positively expansive homeomorphism is finite, Dynamics & stochastics, IMS Lecture Notes Monogr. Ser., vol. 48, Inst. Math. Statist., Beachwood, OH, 2006, pp. 304–305. MR 2306210, DOI 10.1214/074921706000000310
Dawoud Ahmadi Dastjerdi and Maryam Hosseini, Sub-shadowings, Nonlinear Anal. 72 (2010), no. 9-10, 3759–3766. MR 2606819, DOI 10.1016/j.na.2010.01.014
Timo Eirola, Olavi Nevanlinna, and Sergei Yu. Pilyugin, Limit shadowing property, Numer. Funct. Anal. Optim. 18 (1997), no. 1-2, 75–92. MR 1442020, DOI 10.1080/01630569708816748
Abbas Fakhari and F. Helen Ghane, On shadowing: ordinary and ergodic, J. Math. Anal. Appl. 364 (2010), no. 1, 151–155. MR 2576059, DOI 10.1016/j.jmaa.2009.11.004
Leobardo Fernández and Chris Good, Shadowing for induced maps of hyperspaces, Fund. Math. 235 (2016), no. 3, 277–286. MR 3570213, DOI 10.4064/fm136-2-2016
C. Good, S. Greenwood, R. W. Knight, D. W. McIntyre, and S. Watson, Characterizing continuous functions on compact spaces, Adv. Math. 206 (2006), no. 2, 695–728. MR 2263719, DOI 10.1016/j.aim.2005.11.002
Chris Good and Jonathan Meddaugh, Orbital shadowing, internal chain transitivity and $\omega$-limit sets, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 143–154. MR 3742540, DOI 10.1017/etds.2016.30
Chris Good and Jonathan Meddaugh, Shifts of finite type as fundamental objects in the theory of shadowing, Invent. Math. 220 (2020), no. 3, 715–736. MR 4094967, DOI 10.1007/s00222-019-00936-8
Chris Good, Jonathan Meddaugh, and Joel Mitchell, Shadowing, internal chain transitivity and $\alpha$-limit sets, J. Math. Anal. Appl. 491 (2020), no. 1, 124291, 19. MR 4109096, DOI 10.1016/j.jmaa.2020.124291
Chris Good, Joel Mitchell, and Joe Thomas, On inverse shadowing, Dyn. Syst. 35 (2020), no. 3, 539–547. MR 4149093, DOI 10.1080/14689367.2020.1739228
Chris Good, Joel Mitchell, and Joe Thomas, Preservation of shadowing in discrete dynamical systems, J. Math. Anal. Appl. 485 (2020), no. 1, 123767, 39. MR 4043023, DOI 10.1016/j.jmaa.2019.123767
Chris Good, Piotr Oprocha, and Mate Puljiz, Shadowing, asymptotic shadowing and s-limit shadowing, Fund. Math. 244 (2019), no. 3, 287–312. MR 3893427, DOI 10.4064/fm492-5-2018
Marcin Kulczycki and Piotr Oprocha, Properties of dynamical systems with the asymptotic average shadowing property, Fund. Math. 212 (2011), no. 1, 35–52. MR 2771587, DOI 10.4064/fm212-1-3
Keonhee Lee, Continuous inverse shadowing and hyperbolicity, Bull. Austral. Math. Soc. 67 (2003), no. 1, 15–26. MR 1962956, DOI 10.1017/S0004972700033487
Keonhee Lee and Kazuhiro Sakai, Various shadowing properties and their equivalence, Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 533–540. MR 2152404, DOI 10.3934/dcds.2005.13.533
Jie Li and Ruifeng Zhang, Levels of generalized expansiveness, J. Dynam. Differential Equations 29 (2017), no. 3, 877–894. MR 3694815, DOI 10.1007/s10884-015-9502-6
Jonathan Meddaugh and Brian E. Raines, Shadowing and internal chain transitivity, Fund. Math. 222 (2013), no. 3, 279–287. MR 3104074, DOI 10.4064/fm222-3-4
Joel Mitchell, Orbital shadowing, $\omega$-limit sets and minimality, Topology Appl. 268 (2019), 106903, 7. MR 4014335, DOI 10.1016/j.topol.2019.106903
Carlos Morales, A generalization of expansivity, Discrete Contin. Dyn. Syst. 32 (2012), no. 1, 293–301. MR 2837062, DOI 10.3934/dcds.2012.32.293
Helena E. Nusse and James A. Yorke, Is every approximate trajectory of some process near an exact trajectory of a nearby process?, Comm. Math. Phys. 114 (1988), no. 3, 363–379. MR 929137
Piotr Oprocha, Transitivity, two-sided limit shadowing property and dense $\omega$-chaos, J. Korean Math. Soc. 51 (2014), no. 4, 837–851. MR 3225211, DOI 10.4134/JKMS.2014.51.4.837
Piotr Oprocha, Shadowing, thick sets and the Ramsey property, Ergodic Theory Dynam. Systems 36 (2016), no. 5, 1582–1595. MR 3519424, DOI 10.1017/etds.2014.130
D. W. Pearson, Shadowing and prediction of dynamical systems, Math. Comput. Modelling 34 (2001), no. 7-8, 813–820. MR 1858802, DOI 10.1016/S0895-7177(01)00101-7
Timothy Pennings and Jeffrey Van Eeuwen, Pseudo-orbit shadowing on the unit interval, Real Anal. Exchange 16 (1990/91), no. 1, 238–244. MR 1087487
S. Yu. Pilyugin, A. A. Rodionova, and K. Sakai, Orbital and weak shadowing properties, Discrete Contin. Dyn. Syst. 9 (2003), no. 2, 287–308. MR 1952375, DOI 10.3934/dcds.2003.9.287
Sergei Yu. Pilyugin, Shadowing in dynamical systems, Lecture Notes in Mathematics, vol. 1706, Springer-Verlag, Berlin, 1999. MR 1727170
Sergei Yu. Pilyugin, Sets of dynamical systems with various limit shadowing properties, J. Dynam. Differential Equations 19 (2007), no. 3, 747–775. MR 2350246, DOI 10.1007/s10884-007-9073-2
Clark Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), no. 3, 425–437. MR 494300, DOI 10.1216/RMJ-1977-7-3-425
Kazuhiro Sakai, Various shadowing properties for positively expansive maps, Topology Appl. 131 (2003), no. 1, 15–31. MR 1982811, DOI 10.1016/S0166-8641(02)00260-2
Peter 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., vol. 668, Springer, Berlin, 1978, pp. 231–244. MR 518563