Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-04T21:16:20.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamical properties of shift maps on inverse limits with a set valued function

Published online by Cambridge University Press:  22 September 2016

JUDY KENNEDY  and
VAN NALL
JUDY KENNEDY
Affiliation:
Department of Mathematics, PO Box 10047, Lamar University, Beaumont, TX 77710, USA email [email protected]
VAN NALL
Affiliation:
Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173, USA email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 4 , June 2018 , pp. 1499 - 1524
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akin, E.. General Topology of Dynamical Systems (Graduate Studies in Mathematics Series, 1) . American Mathematical Society, Providence, RI, 1993.Google Scholar
Banič, I.. Inverse limits as limits with respect to the Hausdorff metric. Bull. Aust. Math. Soc. 75 (2007), 17–22.Google Scholar
Banič, I., Črepnjak, M., Merhar, M. and Milutinovič, U.. Limits of inverse limits. Topology Appl. 157 (2010), 439–450.Google Scholar
Banič, I., Črepnjak, M., Merhar, M. and Milutinovič, U.. Towards the complete classification of tent maps inverse limits. Topology Appl. 160 (2013), 63–73.Google Scholar
Banič, I., Črepnjak, M., Merhar, M., Milutinovič, U. and Sovič, T.. Ważewski’s universal dendrite as an inverse limit with one set-valued bonding function. Glas. Mat. Ser. III 48 (2013), 137–165.Google Scholar
Banič, I. and Kennedy, J.. Inverse limits with bonding functions whose graphs are arcs. Topology Appl. 151 (2015), 9–21.CrossRefGoogle Scholar
Case, J. H. and Chamberlin, R. E.. Characterizations of tree-like continua. Pacific J. Math. 10 (1960), 73–84.Google Scholar
Mouran, C.. Positive entropy homeomorphisms of chainable continua and indecomposable subcontinua. Proc. Amer. Math. Soc. 139 (2010), 2783–2791.Google Scholar
Charatonik, W. J. and Roe, R. P.. Inverse limits of continua having trivial shape. Houston J. Math. 38(4) (2012), 1307–1312.Google Scholar
Devaney, R.. An Introduction to Chaotic Dynamical Systems, 2nd edn. Westview Press, Cambridge, MA, 2003.Google Scholar
Greenwood, S. and Kennedy, J.. Connected generalized inverse limits. Topology Appl. 159 (2012), 57–68.Google Scholar
Greenwood, S. and Kennedy, J.. Connectedness and Ingram–Mahavier products. Topology Appl. 166 (2014), 1–9.Google Scholar
Illanes, A.. A circle is not the generalized inverse limit of a subset of [0, 1]2 . Proc. Amer. Math. Soc. 139 (2011), 2987–2993.Google Scholar
Ingram, W. T.. Two-pass maps and indecomposability of inverse limits of graphs. Topology Proc. 29 (2005), 1–9.Google Scholar
Ingram, W. T.. Inverse limits of upper semicontinuous functions that are unions of mappings. Topology Proc. 34 (2009), 17–26.Google Scholar
Ingram, W. T.. Inverse limits with upper semicontinuous bonding functions: problems and some partial solutions. Topology Proc. 36 (2010), 353–373.Google Scholar
Ingram, W. T.. Inverse limits of upper semicontinuous set valued functions. Houston J. Math. 32(1) (2006), 17–26.Google Scholar
Ingram, W. T.. An Introduction to Inverse Limits with Set-Valued Functions. Springer, New York, NY, 2012.Google Scholar
Ingram, W. T. and Mahavier, W.. Inverse Limits: From Continua to Chaos. Springer, New York, NY, 2012.Google Scholar
Ingram, W. T. and Mahavier, W. S.. Inverse limits of upper semicontinuous set valued functions. Houston J. Math. 32 (2006), 119–130.Google Scholar
Mahavier, W. S.. Inverse limits with subsets of [0, 1] × [0, 1]. Topology Appl. 141 (2004), 225–231.Google Scholar
Mioduszewski, J.. On a quasi-ordering in the class of continuous mappings of a closed interval into itself. Colloq. Math. 9 (1962), 233–240.Google Scholar
Nall, V.. Connected Inverse limits with set-valued functions. Topology Proc. 40 (2012), 167–177.Google Scholar
Nall, V.. Inverse limits with set valued functions. Houston J. Math. 37(4) (2011), 1323–1332.Google Scholar
Nall, V.. Finite graphs that are inverse limits with a set valued function on [0, 1]. Topology Appl. 158 (2011), 1226–1233.Google Scholar
Nall, V.. The only finite graph that is an inverse limit with a set valued function on [0, 1] is an arc. Topology Appl. 159 (2012), 733–736.Google Scholar
Robinson, C.. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd edn. CRC Press, Boca Raton, FL, 1998.CrossRefGoogle Scholar
Li, S.. Dynamical properties of shift maps on inverse limit spaces. Ergod. Th. & Dynam. Sys. 12 (1992), 95–108.Google Scholar
Varagona, S.. Inverse limits with upper semi-continuous bonding functions and indecomposability. Houston J. Math. 37 (2011), 1017–1034.Google Scholar