LI-YORKE CHAOTIC GENERALIZED SHIFT DYNAMICAL SYSTEMS

Document Type : Research Articles

Authors

¹ Faculty of Math., Stat. and Computer Science, College of Science, University of Tehran, Tehran, Iran

² Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract

‎In this text we prove that in generalized shift dynamical system

(X^{G} a m m a, s i g m a_{v} a r p h i)

‎ ‎for finite discrete

X

with at least two elements‎, ‎infinite countable set

G a m m a

and‎ ‎arbitrary map

v a r p h i : G a m m a t o G a m m a

‎, ‎the following statements are equivalent‎:
‎
- the dynamical system

(X^{G} a m m a, s i g m a_{v} a r p h i)

is‎ Li-Yorke chaotic;‎
- the dynamical system

(X^{G} a m m a, s i g m a_{v} a r p h i)

has‎ an scrambled pair;‎
‎- the map

v a r p h i : G a m m a t o G a m m a

has at least‎ one non-quasi-periodic point‎.

Keywords