Some Kinds of Distributional Chaos for Non-Autonomous Discrete Systems

Corresponding Author: Tianxiu Lu College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, 643000, P.R. China Email: lubeeltx@163.com Abstract: In this study, several kinds of distributional chaos are defined in a non-autonomous discrete system and the chaotic behavior of the mapping sequence , 1 =( , , ) n n n f f f   , n   ( is a set of natural numbers) is studied.


Introduction
The definition of chaos in dynamics was started in 1975 by Li and Yorke (1975). They studied pairs of points with the property that their orbits are neither asymptotic nor separated by any positive fixed constant. To describe the complexity and unpredictability of the system from different perspectives, various definitions of chaos have been proposed, such as Devaney chaos (Banks, 1992), Generic chaos (Snoha, 1990), dense chaos (Snoha et al., 1992), dense δ-chaos (Ruette, 2005), Li-Yorke sensitivity (Akin and Kolyada, 2003) and so on. While, an important extension of Li-Yorke chaos is distributional chaos, which is introduced by Schweizer and Smital (1994). The related concept distributional chaotic pair as two points for which the statistical distribution of distances between the orbits does not converge, and Schweizer and Smital (1994) proved that the existence of a single distributional chaotic pair is equivalent to the positive topological entropy (and some other notions of chaos) when restricted to the compact interval case. Since then, distributional chaos has been widely concerned in dynamical system theory (see Smítal and Štefánková, 2004;Balibrea et al., 2005;Martínez-Giménez et al., 2009;Liao et al., 2009;Oprocha, 2009;Li, 2011;Dvorakova, 2011;Wu and Chen, 2013;Shao et al., 2018). Smítal and Štefánková (2004) showed that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugation. To describe distributional chaos in more detail, distributional chaos of type 1 ( 1 DC ), of type 2 ( 2 DC ), and of type 3 ( 3 DC ) are proposed. Balibrea et al. (2004) showed that DC3 does not imply chaos in the sense of Li and Yorke. They also showed that DC3 is not invariant with respect to topological conjugation. Contrary to this, either 1 DC or 2 DC is topological conjugation invariant and implies Li and Yorke chaos. Then, Martinez-Gimenez (2008) provided sufficient conditions which give uniformly distributional chaos for backward shift operators, and Liao (2009) gave an example which is mixing but not distributively chaotic. In the same year, Oprocha (2009) (Wu and Chen, 2013)).
is an autonomous discrete system ( , ) If .
Next section, several definitions of distributional chaoticity are given. In section 3 and section 4, the main results are established.

Preliminaries
Similar to the definition of distributional chaos in autonomous systems, the following defines distributional chaos in the case of non-autonomous.

Proof
Only the sufficiency of the case 2 n  is proved. The necessity is similar, where, || A represents the cardinality of the set A .
Since 1 f is surjective, then for any ,, Taking an inverse image of each element in S under 1 f , and let T is the set of these inverse images, then T is an uncountable set. The following will prove that T is the distributional scrambled set of 1, In the following proof, ** , , , x y S T are the same as the proof in (1). xy , .
x y for any x, y ∈ S and any 0 tr  , then for any ** , That is to say, 1, x y S  and any 0 t   , then for any ** , (5) For uniformly distributional chaos, by (a), This means that f1, is uniformly distributional chaos. , then for any ** , x y T  and any

Distributional Chaoticity of Compound
System of 1, f  This section mainly discusses the relationship between the distributional chaos of f1, and the distributional chaos of the compound system [] 1, m f  (m is a positive integer). In the following, it is always assumed that fn (n  ℕ) are surjective, which is a common condition when dealing with these systems.
For any m  , denote:    Lemma 4.5  If the mapping sequence The proof of upper distributional function for 1, f  is the same as the proof of uniformly distributional chaos, so it is omitted. The following will lower upper distributional function for 1, f  .
In fact, for any 0 <  < diamI, there exist s: 0 < s <  and N  ℕ such that, for any x, y  I: (x, y) and any k  N ( ( ), ( ))