Hyperbolic dynamics in graph-directed IFS

A cone space   is a complete metric space (X,d)(X,d) with a pair of functions cs,cu:X×X→Rcs,cu:X×X→R, such that there exists K>0K>0 satisfying1Kd(x,x′)⩽max(cs(x,x′),cu(x,x′))⩽Kd(x,x′)forx,x′∈X. For a partial map f between cone spaces X and Y   we introduce |f|s|f|s which measures the stable contraction rate and 〈f〉u〈f〉u which measures the unstable expansion rate. We say that f is cone-hyperbolic if|f|s<1<〈f〉u.|f|s<1<〈f〉u.Using cone field and graph-directed IFS we build an abstract metric model which describes the dynamics of the hyperbolic-like systems. This allows to obtain estimations from below and above of the fractal dimension of the hyperbolic invariant set.