On sensitivity and transitivity of a dynamical system and its induced dynamical systems

Abstract

Let ( X , f ) be a dynamical system, i.e. X is a compact metric space and f is a continuous self-map on X and let K ( X ) , M ( X ) and F ( X ) be the sets of all non-empty compact subsets of X, Borel probability measures on X and upper semi-continuous fuzzy sets on X, respectively. Then K ( X ) , M ( X ) and F ( X ) are metric spaces under Hausdorff metric, prohorov metric and level-wise metric, respectively. Therefore, ( X , f ) naturally induces three new systems ( K ( X ) , f ¯ ) , ( M ( X ) , f ˆ ) and ( F ( X ) , f ~ ) . In this article, we investigate the connection of ( r , s ) -sensitivity, ( r , s ) -asymptotic sensitivity, ( r , s ) -Li-Yorke sensitivity and Δ-transitivity of ( X , f ) and its induced systems ( K ( X ) , f ¯ ) , ( M ( X ) , f ˆ ) and ( F ( X ) , f ~ ) and we obtain some desired results. For instance, we prove that ( K ( X ) , f ¯ ) is ( r , s ) -sensitive ⇔ ( F 1 ( X ) , f g ~ ) is ( r , s ) -sensitive for each g ∈ D m ( I ) satisfying g − 1 ( 1 ) = { 1 } ; ( X , f ) is Δ-transitive ⇔ ( K ( X ) , f ¯ ) is Δ-transitive ⇔ ( M ( X ) , f ˆ ) is Δ-transitive ⇔ ( F 1 ( X ) , f ~ ) is Δ-transitive.