Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi compactness General Facts About The Method Purpose Of The Paper The Central Limit Theorems For Markov Chains Theorems A B C Quasi Compact Operators of Diagonal Type And Their Perturbations First Properties of Fourier Kernels Application Peripheral Eigenvalues of Fourier Kernels Proofs Of Theorems A B C Renewal Theorem For Markov Chains Theorem D Large Deviations For Markov Chains Theorem E Ergodic Properties For Markov Chains Markov Chains Associated With Lipschitz Kernels Examples Stochastic Properties Of Dynamical Systems Theorems A* B* C* D* E* Expanding Maps Proofs Of Some Statements In Probability Theory Functional Analysis Results On Quasi Compactness Generalization To The Non Ergodic Case. The usefulness of from the of techniques perturbation theory operators to kernel for limit theorems for a applied quasi compact positive Q obtaining Markov chains for stochastic of or dynamical by describing properties systems of Perron Frobenius has been demonstrated in several All use a operator papers. these works share the features the features that must be same specific general ; used in each stem from the nature of the functional particular case precise space where the of is and from the number of quasi compactness Q proved eigenvalues of of modulus 1. We here a functional framework for Q give general analytical this method and we the aforementioned behaviour within it. It asymptotic prove is worth that this framework is to allow the unified noticing sufficiently general treatment of all the cases considered in the literature the previously specific ; characters of model translate into the verification of of simple hypotheses every a functional nature. When to Markov kernels or to Perr applied Lipschitz Frobenius associated with these statements rise operators expanding give maps to new results and the of known The main clarify proofs already properties. of the deals with a Markov kernel for which 1 is a part quasi compact Q paper of modulus 1. An essential but is not the simple eigenvalue unique eigenvalue element of the work is the of the of peripheral Q precise description spectrums and of its To conclude the the results obtained perturbations.