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¡¡¡¡this book surveys£¬ in a popular way£¬ the main progress made in the field by our group. it consists of ten chapters plus two appendixes. the first chapter is an overview of the second to the eighth ones. mainly£¬ we study several different inequalities or different types of convergence by using three mathematical tools: a probabilistic tool£¬ the coupling methods £¨chapters 2 and 3£©; a generalized cheeger's method originating in riemannian geometry £¨chapter 4£©; and an approach coming from potential theory and harmonic analysis £¨chapters 6 and 7£©. the explicit criteria for different types of convergence and the explicit estimates of the convergence rates £¨or the optimal constants in the inequalities£© in dimension one are given in chapters 5 and 6; some generalizations are given in chapter 7. the proofs of a diagram of nine types of ergodicity £¨theorem 1.9£© are presented in chapter 8.