Attractors Hierarchy For Axiom A Diffeomorphisms And The Symbolic Dynamics Of Tel Mapping

In part one, we discuss the attractors hierarchy for axiom A diffeomorphisms. First, we describe the Birkhoff center, the minimal attractive center and the minimal attractor. Second, we give relationships among the attractor of axiom A, the non-wandering set, the limit set, the Birkhoff center, the probability limit set, the minimal attractive center, the minimal attractor, the Ruelle attractor and the measure center.In part two, we discuss the symbolic dynamics of Tel mapping. We define the map n from plane to symbolic space, and define its inverse map x using Milnor's operator. We show that the symbol sequenceswith same tail (resp. head ) will be mapped to the same line under the map ,. From above, we define the pruning front and the primarypruned region mathematically, and give a proof of the pruning front conjecture. Finally, we present some open problem for further study.