# Problem 99

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(Doug (Lind?)) Find open partitions in $\Sigma _{1/2,1/2}$ that are not weakly Bernoulli. Find invariants of finitary codes.

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### Perhaps related to [1] or [2]

Perhaps related to [1]?

### References

1. [bowen1975smooth] Bowen R.  1975.  Israel Journal of Mathematics. 21:95–100.

### Keane and Smorodinsky showed

Keane and Smorodinsky showed that two aperiodic shifts of finite type are finitarily isomorphic iff they have the same topological entropy ("The finitary isomorphism theorem for Markov shifts", Bull. AMS 1 (1979)m 436-438). One consequence is that an ergodic (algebraic) automorphism of the 2-torus has a partition into sets with nonempty interior and bourdary of measure zero that is an independent Bernoulli generator. The reference above to Bowen's paper shows that in dimension 3 such a partition cannot be smooth.