OK - so you are talking about the ergodicity of a Markov chain with respect to a finite stationary measure. One general result you should be aware of is that in this situation ergodicity of the time shift in the path space (this is essentially the definition you use - you just refer to the corresponding ergodic theorem) is equivalent to "irreducibility" (absence of non-trivial invariant subsets) of the state space, see Different uses of the word "ergodic"

However it does not help much. The point is that the question you ask is essentially the same as asking about a "general method" to decide whether a given probability measure preserving transformation is ergodic. For, any such transformation can be considered as a "deterministic" Markov chain (all transition probabilities are delta measures). On the other hand, as I have just mentioned, Markov chain ergodicity is equivalent to ergodicity of the deterministic time shift on its path space.

Now, there are no general results in ergodic theory which, to quote your question "can help to easily establish ergodicity". Indeed, any textbook in ergodic theory opens with the standard set of examples (irrational rotation, Bernoulli shift, Gauss transformation, hyperbolic toral endomorphism), ergodicity of each of which is established in its own way. Unfortunately, these approaches are very far from being universal, and ergodicity of certain well-known transformations is a very hard problem (the most famous example being, arguably, Boltzmann's ergodic hypothesis).

EDIT (additional comments) Harris recurrence is in a sense an artifact which does not have anything to do with ergodicity. It appeared at a very early stage of the theory (its predecessor is Doeblin's condition) and was geared at establishing (fast) convergence in total variation to a stationary distribution. Therefore, absolute continuity of transition probabilities (understood literally or in a somewhat weaker form) was a necessary ingredient of this setup. Markov chains with singular transition probabilities (in particular, deterministic Markov chains corresponding to measure preserving transformations) were in this probabilistic context considered as something utterly exotic. I would recommend the books by Foguel MR0261686 (in the first place) and Rosenblatt MR0329037 (more specific) for an "ergodic theory" approach to Markov chains. Both these books contain the statement I had mentioned.

OK - so you are talking about the ergodicity of a Markov chain with respect to a finite stationary measure. One general result you should be aware of is that in this situation ergodicity of the time shift in the path space (this is essentially the definition you use - you just refer to the corresponding ergodic theorem) is equivalent to "irreducibility" (absence of non-trivial invariant subsets) of the state space, see Different uses of the word "ergodic"

However it does not help much. The point is that the question you ask is essentially the same as asking about a "general method" to decide whether a given probability measure preserving transformation is ergodic. For, any such transformation can be considered as a "deterministic" Markov chain (all transition probabilities are delta measures). On the other hand, as I have just mentioned, Markov chain ergodicity is equivalent to ergodicity of the deterministic time shift on its path space.

Now, there are no general results in ergodic theory which, to quote your question "can help to easily establish ergodicity". Indeed, any textbook in ergodic theory opens with the standard set of examples (irrational rotation, Bernoulli shift, Gauss transformation, hyperbolic toral endomorphism), ergodicity of each of which is established in its own way. Unfortunately, these approaches are very far from being universal, and ergodicity of certain well-known transformations is a very hard problem (the most famous example being, arguably, Boltzmann's ergodic hypothesis).

EDIT (additional comments) Harris recurrence is in a sense an artifact which does not have anything to do with ergodicity. It appeared at a very early stage of the theory (its predecessor is Doeblin's condition) and was geared at establishing (fast) convergence in total variation to a stationary distribution. Therefore, absolute continuity of transition probabilities (understood literally or in a somewhat weaker form) was a necessary ingredient of this setup. Markov chains with singular transition probabilities (in particular, deterministic Markov chains corresponding to measure preserving transformations) were in this probabilistic context considered as something utterly exotic. I would recommend the books by Foguel MR0261686 (in the first place) and Rosenblatt MR0329037 (more specific) for an "ergodic theory" approach to Markov chains. Both these books contain the statement I had mentioned.

OK - so you are talking about the ergodicity of a Markov chain with respect to a finite stationary measure. One general result you should be aware of is that in this situation ergodicity of the time shift in the path space (this is essentially the definition you use - you just refer to the corresponding ergodic theorem) is equivalent to "irreducibility" (absence of non-trivial invariant subsets) of the state space, see Different uses of the word "ergodic"

However it does not help much. The point is that the question you ask is essentially the same as asking about a "general method" to decide whether a given probability measure preserving transformation is ergodic. For, any such transformation can be considered as a "deterministic" Markov chain (all transition probabilities are delta measures). On the other hand, as I have just mentioned, Markov chain ergodicity is equivalent to ergodicity of the deterministic time shift on its path space.

Now, there are no general results in ergodic theory which, to quote your question "can help to easily establish ergodicity". Indeed, any textbook in ergodic theory opens with the standard set of examples (irrational rotation, Bernoulli shift, Gauss transformation, hyperbolic toral endomorphism), ergodicity of each of which is established in its own way. Unfortunately, these approaches are very far from being universal, and ergodicity of certain well-known transformations is a very hard problem (the most famous example being, arguably, Boltzmann's ergodic hypothesis).

OK - so you are talking about the ergodicity of a Markov chain with respect to a finite stationary measure. One general result you should be aware of is that in this situation ergodicity of the time shift in the path space (this is essentially the definition you use - you just refer to the corresponding ergodic theorem) is equivalent to "irreducibility" (absence of non-trivial invariant subsets) of the state space, see Different uses of the word "ergodic"

However it does not help much. The point is that the question you ask is essentially the same as asking about a "general method" to decide whether a given probability measure preserving transformation is ergodic. For, any such transformation can be considered as a "deterministic" Markov chain (all transition probabilities are delta measures). On the other hand, as I have just mentioned, Markov chain ergodicity is equivalent to ergodicity of the deterministic time shift on its path space.

Now, there are no general results in ergodic theory which, to quote your question "can help to easily establish ergodicity". Indeed, any textbook in ergodic theory opens with the standard set of examples (irrational rotation, Bernoulli shift, Gauss transformation, hyperbolic toral endomorphism), ergodicity of each of which is established in its own way. Unfortunately, these approaches are very far from being universal, and ergodicity of certain well-known transformations is a very hard problem (the most famous example being, arguably, Boltzmann's ergodic hypothesis).

EDIT (additional comments) Harris recurrence is in a sense an artifact which does not have anything to do with ergodicity. It appeared at a very early stage of the theory (its predecessor is Doeblin's condition) and was geared at establishing (fast) convergence in total variation to a stationary distribution. Therefore, absolute continuity of transition probabilities (understood literally or in a somewhat weaker form) was a necessary ingredient of this setup. Markov chains with singular transition probabilities (in particular, deterministic Markov chains corresponding to measure preserving transformations) were in this probabilistic context considered as something utterly exotic. I would recommend the books by Foguel MR0261686 (in the first place) and Rosenblatt MR0329037 (more specific) for an "ergodic theory" approach to Markov chains. Both these books contain the statement I had mentioned.