Dynamics Of The Induced Maps On The Inverse Limit Spaces

For each i ∈N, let (Xi, di) be a compact metric space. Suppose that fi:Xi+1 →Xi and gi : Xi→Xi are continuous maps satisfying giofi = fiogi+1. Let X∞ be the inverse limit of the maps {fi}i=1∞ and g∞ be the map induced by {gi}i=1∞ on the inverse limit space X∞.In this paper, we mainly discuss the dynamical properties of the induced map g∞ on the inverse limit space X∞ including weak specification and uniform positive entropy. We present sufficient and necessary conditions for g∞ to be equicontinuous(resp. uniformly rigid, mild mixing), and show a sufficient condition for g∞ to be complete chaos.Our main results are the following:(1)g∞ satisfies weak specification if and only if for each i∈N, gi satisfies weak specification.(2)g∞ has n-uniform positive entropy if and only if for each i∈N, gi has n-uniform positive entropy.(3)g∞ is equicontinuous if and only if for each i ∈N, gi is equicontinuous .(4) g∞ is uniformly rigid if and only if for each i ∈N, gi is uniformly rigid.(5) g∞ is mild mixing if and only if for each i ∈N, gi is mild mixing.(6)if for each i ∈ N, gi is complete chaos, then g∞ is complete chaos. However, the converse is not true.