Nan Jiang - School of Science Xi&#۰۳۹;an Shiyou University Xi&#۰۳۹;an, China
Wei Li - Institute for Advanced Studies in The History of Science Northwest University Xi&#۰۳۹;an, China
Fei Li - School of Science Xi&#۰۳۹;an Shiyou University Xi&#۰۳۹;an, China
Juntao Wang - School of Science Xi&#۰۳۹;an Shiyou University Xi&#۰۳۹;an, China

As the research object of modern nonlinear science‎, ‎a fractal theory has been an important research‎ ‎content for scholars since it comes into the world‎. ‎Moreover‎, ‎iterated function system (IFS) is a significant research object of fractal theory‎. ‎On the other hand‎, ‎the Markov process plays an important role in the stochastic process‎. ‎In this paper‎, ‎the iterated function system with probability(IFSP) and the infinite function system with‎ ‎probability(IIFSP) are investigated by using interlink‎, ‎period‎, ‎recurrence and some related concepts‎. ‎Then‎, ‎some important properties are obtained‎, ‎such as‎: ‎۱‎. ‎The sequence of stochastic variable \{\zeta_{n},(n\geq ۰)\}‎ ‎is a homogenous Markov chain‎. ‎۲‎. ‎The sequence of stochastic variable \{\zeta_{n},(n\geq ۰)\} is an irreducible ergodic chain‎. ‎۳‎. ‎The distribution of transition probability p_{ij}^{(n)} based on n\rightarrow\infty is a stationary probability distribution‎. ‎۴‎. ‎The state space can be decomposed of the union of the finite(or countable) mutually disjoint subsets‎, ‎which are composed of non-recurrence states and recurrence states respectively‎.

Fractal‎, ‎Markov process‎, ‎Iterated function system‎, ‎Probability‎