Suggested Exercises 4:
1. Show that for an integer k, h(T k ) = |k|h(T ) (allow k < 0 only if T is invertible); in
particular, if T is invertible, h(T ) = h(T −1 ).
2. Show that for an irrational rotation T of the circle and α a partition into two semicircles, α0n−1 has at most 2n elements. Use this to directly prove that h(T, α) = 0.
3. Let X and Y be stationary processes. Let T + and S + be the MPT’s corresponding to
the one-sided versions of X and Y . Let T and S be the MPT’s corresponding to the
two-sided versions of X and Y . Show that if T + and S + are isomorphic, then T and
S are isomorphic.
4. Let T + and S + be the MPT’s corresponding to the one-sided versions of iid(p) and
iid(q), where p and q are strictly positive probability vectors, not necessarily of the
same length. Show that T + and S + are isomorphic iff p and q are the same, up to
a reordering (in particular, they have the same length). Deduce that that for MPT’s
corresponding to iid processes, one-sided isomoprhism is strictly stronger than twosided isomorphism.
5. Show that an ergodic MPT T has zero entropy iff it is invertible and has a onesided generator (you may assume that the underlying measure space (X, A, µ) has the
property that any bijective self-mapping of the measure algebra  is induced by a
unique (up to measure zero) invertible measure-preserving transformation; here, Â is
the set of equivalence classes of elements of A with respect to the equivalence relation
A ∼ B iff µ(A∆B) = 0).
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