Resumen de N-Convergence and Chaotic Properties of Non-autonomous Discrete Systems

Risong Li, Michal Málek

• Let f1,∞ := ( fn)∞ n=1 be a non-autonomous dynamical system on a compact metric space X. For a given N ∈ N we consider Nth iterate f [N] 1,∞ of the system (i.e. f [N] 1,∞ = ( f N N(n−1)+1)∞ n=1, where f n i = fi+(n−1)◦···◦ fi and f 0 1 = idX .) We also investigate Nconvergent non-autonomous systems this is weaker notion than uniform convergence.

  In this setting we generalize results regarding different types of chaos. Particularly we prove (1) f1,∞ is distributionally chaotic of type 1 if and only if f [N] 1,∞ is also.

  (2) f1,∞ is distributionally chaotic of type 2 if and only if f [N] 1,∞ is also.

  (3) f1,∞ is distributionally chaotic of type 2 1 2 if and only if f [N] 1,∞ is also.

  (4) f1,∞ is P-chaotic if and only if f [N] 1,∞ is also, where P-chaos denotes one of the following properties: Li-Yorke chaos, dense chaos, dense δ-chaos, generic chaos, generic δ-chaos, Li-Yorke sensitivity and spatio-temporal chaos.

  (5) f1,∞ is sensitive (resp. ergodically sensitive) if and only if f [N] 1,∞ is also.

  We also discuss and partly solve a problem given by [Xinxing Wu, Peiyong Zhu, Chaos in a class of non-autonomous discrete systems. Applied Mathematics Letters 26 (2013) 431-436]. Furthermore, we present two examples which show that conditions of N-convergence and continuity in some results cannot be removed.