Nonwandering set of points of skew-product maps with base having closed set of periodic points

The aim of the present paper is to study the structure of the nonwandering set of points Ω ( ⋅ ) for the skew-product maps C Δ ∗ ( I 2 ) of the unit square I 2 = [ 0 , 1 ] × [ 0 , 1 ] , ( x , y ) → ( f ( x ) , g ( x , y ) ) , with base f having closed set of periodic points. For every F ∈ C Δ ∗ ( I 2 ) and every point ( x , y ) with x periodic of period p x by f and y not chain recurrent of F p x | I x , where I x = { x } × I , we prove that ( x , y ) ∉ Ω ( F ) . On the other hand we construct a map F 0 ∈ C Δ ∗ ( I 2 ) with an isolated fixed point x 0 of f and y 0 ∉ Ω ( F | I x 0 ) such that ( x 0 , y 0 ) ∈ Ω ( F 0 ) .