Li-Yorke sensitivity

By Ethan Akin, S. Kolyada. 2003

254 citations recorded by the source corpus.

We introduce and study a concept which links the Li–Yorke versions of chaos with the notion of sensitivity to initial conditions. We say that a dynamical system (X,T) is Li–Yorke sensitive if there exists a positive e such that every xX is a limit of points yX such that the pair (x,y) is proximal but not e-asymptotic, i.e. for infinitely many positive integers i the distance ρ(Ti(x),Ti(y)) is greater than e but for any positive δ this distance is less than δ for infinitely many i.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d8eeee3310b85f760dfe5c42b86da97d0d7770ac

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In Computational Theory and Mathematics

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Citations in the corpus

Measure-theoretical sensitivity and equicontinuity (2011, background): “In [ 1 ] Akin and Kolyada introduced a notion called Li-Yorke sensitivity, which is much stronger than sensitivity, and showed that a weakly mixing t.d.s.”
Stronger forms of sensitivity for dynamical systems (2007, background): “[ 3 ]. Li-Yorke sensitivity is formulated in terms of special type of individual points in the system.”
Analogues of Auslander–Yorke theorems for multi-sensitivity (2015, background): “exists δ>0 such that for every x∈ X and every neighborhood Ux of x, there exist y ∈ Ux and n∈ Nwith (Tn(x),Tn(y)) >δ, where Nis the set of all natural numbers (positive integers). According to [4], it is easy to see that (X,T) is sensitive if and only if ST(U,δ) is inﬁnite for some δ>0 and ”
Recent development of chaos theory in topological dynamics (2015, background): “residual in X. Later it was proved by Furstenberg in [26] that the proximal cell of every point is residual, provided that the system is minimal and weakly mixing. Finally, Akin and Kolyada proved in [6] that in any weakly mixing system the proximal cell of every point is residual. Note that the aut”
DISTRIBUTIONAL CHAOS REVISITED (2009, background): “We define a map g : [-1, 1] x B -> [-1, 1] x B by putting g(x, y, z) = (x, /(» , 2)).”
Li-Yorke chaos in the system with impacts (2009, background): “The following definitions are taken from [5,18,24] and adapted for (1).”