Topological And Metric Structures Of Peirodic Orbits

This paper discusses the discrete dynamical system problems for a continuousmap on a line, summarized the proving method of Sarkovskii theorem. In the prooffor Sarkovskii theorem, there are some periodic variation relations for a periodicpoints between original map and its multiple composite map. It gives several unifiedconclusions and specific practical inference for the periodic variation relations. Itdiscusses the geometric number and algebraic number of periodic points for period p,gives the geometric number of periodic points, and estimates the upper and lowerbounds of the algebraic number. It analyses the metric structure of periodic orbits,puts forward the concept of multiple periodic orbit, gives a detailed analysis for thedouble periodic orbit with period2, proves that at this time the mutiples of periodicpoints in the period orbit must be equal, hence the same multiple can be defined as themultiple for the periodic orbit. Obtains the result, the necessary and sufficientcondition for double periodic orbit is that the orbit characteristic is1. And it gives theorbit characteristic by the second derivative, and gives a sufficient condition for thetriple periodic orbits. At last, it analyses the topology structure of period orbits. Itintroduces the topological graph representation and the matrix representation forperiodic orbits, obtains the classification topologically for period orbits, and provesthe class number is (p-1)! for the period orbit with period p, specifically draws thetopology graph for the period orbit with period2,3,4and5.