Transitive sensitive subsystems for interval maps

Sylvie Ruette
2005 Studia Mathematica  
We prove that for continuous interval maps the existence of a non-empty closed invariant subset which is transitive and sensitive to initial conditions is implied by positive topological entropy and implies chaos in the sense of Li-Yorke, and we exhibit examples showing that these three notions are distinct. [81] Recently, Blanchard, Glasner, Kolyada and Maass [4] proved that, if T : X → X is a continuous map on the compact metric space X such that the topological entropy of T is positive, then
more » ... the system is chaotic in the sense of Li-Yorke. The converse of this result is not true, even for interval maps: Smítal [23] and Xiong [25] built interval maps of zero entropy which are chaotic in the sense of Li-Yorke. See also [20] (a correction is given in [18]) or [11] for examples of C ∞ interval maps which are chaotic in the sense of Li-Yorke and have a null entropy. Recall that the map T : X → X is transitive if for all non-empty open subsets U, V there exists an integer n ≥ 0 such that T −n (U ) ∩ V = ∅; if X is compact with no isolated point, then T is transitive if and only if there exists x ∈ X such that ω(x, T ) = X (where ω(x, T ) is the set of limit points of {T n (x) | n ≥ 0}). The map T has sensitive dependence on initial conditions (or simply is sensitive) if there exists δ > 0 such that for all x ∈ X and all neighbourhoods U of x there exist y ∈ U and n ≥ 0 such that The work of Wiggins [24] leads to the following definition (see, e.g., [13] ). Definition 1.3. Let X be a metric space. The continuous map T : X → X is said to be chaotic in the sense of Wiggins if there exists a nonempty closed invariant subset Y such that the restriction T | Y is transitive and sensitive.
doi:10.4064/sm169-1-6 fatcat:4ntuvkybnjgpzaunsckodzu4na