| Summary: | 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: 1. The sequence of stochastic variable $\{\zeta_{n},(n\geq 0)\}$ is a homogenous Markov chain. 2. The sequence of stochastic variable $\{\zeta_{n},(n\geq 0)\}$ is an irreducible ergodic chain. 3. The distribution of transition probability $ p_{ij}^{(n)}$ based on $n\rightarrow\infty $ is a stationary probability distribution. 4. 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.
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