Acta mathematica scientia,Series B ›› 2014, Vol. 34 ›› Issue (5): 1473-1480.doi: 10.1016/S0252-9602(14)60097-7

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ON DISTRIBUTIONAL n-CHAOS

 TAN Feng, FU He-Man   

  1. School of the mathematical science, South China Normal University, Guangzhou 510631, China； College of Mathematics and Information Sciences, Zhaoqing University, Zhaoqing 526061, China
  • Received:2012-04-08 Revised:2014-06-12 Online:2014-09-20 Published:2014-09-20
  • Supported by:

    The authors are supported by the NNSF of China (11071084, 11026095 and 11201157), and by FDYT of Guangdong Province (2012LYM 0133).

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Abstract:

Let (X, f) be a topological dynamical system, where X is a nonempty compact and metrizable space with the metric d and f : X → X is a continuous map. For any integer n ≥ 2, denote the product space by X(n) = X × · · · × X. We say a system (X, f) is generally distributionally n-chaotic if there exists a residual set D ⊂X(n) such that for any point x = (x1, · · · , xn) ∈ D,
liminfk→∞#({i : 0 ≤ i ≤ k − 1,min{d(f i(xj), f i(xl)) : 1 ≤ j ≠ l ≤ n} < δ0})/k= 0
for some real number δ0 > 0 and

limsupfk→∞#({i : 0 ≤ i ≤ k − 1,max{d(f i(xj), f i(xl)) : 1 ≤ j ≠ l ≤ n} < δ0})/k= 1

for any real number δ > 0, where #(·) means the cardinality of a set. In this paper, we show that for each integer n ≥ 2, there exists a system (X, σ) which satisfies the following conditions: (1) (X, σ) is transitive; (2) (X, σ) is generally distributionally n-chaotic, but has no distributionally (n + 1)-tuples; (3) the topological entropy of (X, σ) is zero and it has an
IT-tuple.

Key words: transitive systems, distributional chaos

CLC Number: 

  • 37B10
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