Ergodic Theory and Dynamical Systems

Research Article

Dynamical properties of the shift maps on the inverse limit spaces

Shihai Lia1

a1 Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

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.

(Received December 01 1990)