Solenoidal automorphisms with specifications

The specification property for solenoidal automorphisms is discussed; a solenoidal automorphism satisfies specification iff is expansive, and satisfies weak specification but not specification iff is ergodic and central spin. These are problems set up byK. Sigmund for homeomorphisms with specification. The proofs for toral case are given byD. Lind. For solenoidal case, a key ingredient in our proofs is splitting theorems on solenoidal groups with respect to described in § 2. Moreover, the following is proved: (i) If obeys specification then satisfies weak specification and is densely periodic. But the converse is not necessarily true. (ii) Every solenoidal automorphism with specification admits a Markov partition. (iii) Every ergodic solenoidal automorphism without specification does not admit Markov partitions. (iv) There exists an expansive homeomorphism with specification which has not Markov partitions.