The Minimally Non-hyperbolic Expanding Set In One-dimensional Dynamic

In this paper,we will discuss that the minimally non-hyperbolic expanding set in C~2 one-dimensional dynamics has three conditions as follows:neutral periodic orbit,periodic sinks or the dynamics that is topologically conjugate to an irrational rotation on S~1.The non-hyperbolic expanding set is charactered by such a conclusion.Let f be a C~2 map of the circle or the interval into itself and f is not topologically conjugate to an irrational rotation.LetΛbe a non-empty compact invariant set.We get the result thatΛis a hyperbolic expanding set if and only if the following conditions are satisfied:·Λdosen't contain critical point,that is,the point z with f'(x)=0;·every periodic point,if there is,is hyperbolic expanding one;this result is Ma(?)é's classic conclusion.The subject of the paper will give the equivalent theorem to Ma(?)é's theorem,and also give the other analytic methods.In the paper,we also focus on Liao's classification of the minimally non-hyperbolic set of f,the application of distortion preperty and the technology of cutting the natural number into sections.We lead in the pre-orbit space to prove the main proposition.At last,we will extend Denjoy's theorem,and make it more useful in any other fields.