Skip to main content
×
Home
    • Aa
    • Aa

Dynamical properties of the shift maps on the inverse limit spaces

  • Shihai Li (a1)
Abstract
Abstract

In this paper, we prove the following results about the shift map on the inverse limit space of a compact metric space and a sole bonding map: (1) the chain recurrent set of the shift map equals the inverse limit space of the chain recurrent set of the sole bonding map. Similar results are proved for the nonwandering set, ω-limit set, recurrent set, and almost periodic set. (2) The shift map on the inverse limit space is chaotic in the sense of Devaney if and only if its sole bonding map is chaotic. (3) If the sole bonding map is ω-chaotic, then the shift map on the inverse limit space is ω-chaotic. With a modification of the definition of ω-chaos we show the converse is true. (4) We prove that a transitive map on a closed invariant set containing an interval must have sensitive dependence on initial conditions on the whole set. Then it is both ω-chaotic and chaotic in the sense of Devaney with chaotic set the whole set. At last we give a new method of constructing chaotic homeomorphisms on chainable continua, especially on pseudo-arcs.

Copyright
References
Hide All
[Ba]Barge M.. A method for constructing attractors. Ergod. Th. & Dynam. Sys. 8 (1988), 331349.
[BaM]Barge M. & Martin J.. Chaos, periodicity, and snakelike continua. Trans. Amer. Math. Soc. 289 (1985), 355365.
[Bir]Birkhoff G.. Dynamical systems. Amer. Math. Soc: New York, 1927.
[Bl]Block L.. Diffeomorphisms obtained from endomorphisms. Trans. Amer. Math. Soc. 214 (1975), 403413.
[BlCa]Block L. & Coven E. M., ω-limit sets for maps of the interval. Ergod. Th. & Dynam. Sys. 6 (1986), 335344.
[BlCb]Block L. & Coven E. M.. Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval. Trans. Amer. Math. Soc. 300 (1987), 297306.
[Blh]Blokh A.. On sensitive mappings of the interval. Russian Math. Surveys 37 (1982), 203204.
[D]Devaney R.. An Introduction to Chaotic Dynamical Systems. The Benjamin/Cummings Publishing Co. Inc.: California, 1986.
[H]Henderson G. W.. The pseudo-arc as an inverse limit with one binding map. Duke Math. J. 31 (1964), 421425.
[K]Kennedy J.. The construction of chaotic homeomorphisms on chainable continua. Trans. Amer. Math. Soc. to appear.
[L]Li S.-H.. ω-chaos and topological entropy. Trans. Amer. Math. Soc. to appear.
[M]Misiurewicz M.. Horseshoes for continuous mappings of an interval. Bull. Acad. Pol. Sci. 27 (1979), 167169.
[MT]Minc P. & Transue W. R. R.. A transitive map on [0, 1] whose inverse limit is the pseudo-arc. Preprint.
[N]Nitecki Z.. Topological dynamics on the interval. Ergod. Th. Dynam. Sys. II Proc. Special Year, Maryland 19791980 (Katok A., ed), Birkhauser: Basel (1982), 173.
[S]Sharkovskii A. N.. On the properties of discrete dynamical systems. Proc. Int. Colloq. on Iteration Theory and its Applications, Toulouse (1982).
[X]Xiong J.-C.. inline-graphic for every continuous self map f of the interval. Kexue Tongbao 28 (1983), 2123 (English version).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 19 *
Loading metrics...

Abstract views

Total abstract views: 149 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.