Strong cocycle triviality for Z2 subshifts

We consider the cohomology of two-dimensional subshifts, and develop a new approach to proving that every cocycle is trivial (i.e. cohomologous to a homomorphism). We introduce semi-safe subshifts which, roughly speaking, have the property that some symbol can surround all allowed blocks in at least one horizontal and one vertical direction. We prove that for such subshifts, every locally constant cocycle with values in a locally (residually nite) group is trivial. Several authors have noted that, in contrast to the situation for Z-actions, the rst cohomology of a Z d-action, for d > 1, can be very small. This is one of several rigidity properties enjoyed by higher dimensional actions on compact sets X. Kammeyer 4], 5], 6] proved that every continuous cocycle on the two-dimensional full shift on n symbols, with values in a nite group, is trivial. Triviality here is in the strong sense that all cocycles are cohomologous to a homomorphism (a cocycle generated by a function constant on the shift space). Schmidt 16] subsequently gave general conditions on a multi-dimensional subshift to ensure the triviality of all cocycles of summable variation taking values in a group with a doubly-invariant metric. Roughly, these conditions are that the subshift should be mixing, and satisfy a certain speciication property. Katok & Schmidt 8] showed that if a mixing subshift X itself carries an abelian group structure (the motivating examples are of the type introduced by Ledrappier 10]) then all real-valued HH older cocycles are strongly trivial in the above sense. In a more geometric context, Katok & Spatzier 7] proved that for certain multi-dimensional Anosov actions on a compact manifold, all real-valued HH older cocycles are again strongly trivial. This strong triviality is not universal (see the examples in 16]), but nonetheless some form of cohomological triviality seems to be a prevalent feature of higher rank actions. If X is a mixing subshift with an abelian group structure, then Schmidt 17] shows that any cocycle taking values in S 1 is cohomologous to an aane cocycle (i.e. one which is the sum of a homomorphism Z d ! S 1 and a homomorphism X ! S 1). Parry 11] gives suucient conditions for the analogous result to hold if the cocycle takes values in a nite abelian group. In Schmidt 18] it is shown that certain mixing subshifts of nite type have a `fundamental' cocycle, which essentially determines all …