3,625 research outputs found
On the Lebesgue measure of Li-Yorke pairs for interval maps
We investigate the prevalence of Li-Yorke pairs for and
multimodal maps with non-flat critical points. We show that every
measurable scrambled set has zero Lebesgue measure and that all strongly
wandering sets have zero Lebesgue measure, as does the set of pairs of
asymptotic (but not asymptotically periodic) points.
If is topologically mixing and has no Cantor attractor, then typical
(w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally
admits an absolutely continuous invariant probability measure (acip), then
typical pairs have a dense orbit for . These results make use of
so-called nice neighborhoods of the critical set of general multimodal maps,
and hence uniformly expanding Markov induced maps, the existence of either is
proved in this paper as well.
For the setting where has a Cantor attractor, we present a trichotomy
explaining when the set of Li-Yorke pairs and distal pairs have positive
two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure
Li-Yorke chaos in hybrid systems on a time scale
By using the reduction technique to impulsive differential equations [1], we
rigorously prove the presence of chaos in dynamic equations on time scales
(DETS). The results of the present study are based on the Li-Yorke definition
of chaos. This is the first time in the literature that chaos is obtained for
DETS. An illustrative example is presented by means of a Duffing equation on a
time scale.Comment: 16 pages, 2 figure
Chaotic Properties of Subshifts Generated by a Non-Periodic Recurrent Orbit
The chaotic properties of some subshift maps are investigated. These
subshifts are the orbit closures of certain non-periodic recurrent points of a
shift map. We first provide a review of basic concepts for dynamics of
continuous maps in metric spaces. These concepts include nonwandering point,
recurrent point, eventually periodic point, scrambled set, sensitive dependence
on initial conditions, Robinson chaos, and topological entropy. Next we review
the notion of shift maps and subshifts. Then we show that the one-sided
subshifts generated by a non-periodic recurrent point are chaotic in the sense
of Robinson. Moreover, we show that such a subshift has an infinite scrambled
set if it has a periodic point. Finally, we give some examples and discuss the
topological entropy of these subshifts, and present two open problems on the
dynamics of subshifts
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