Conjugacy between topology and measure theory
a. Weakest notion such that h(f) is an invariant
b. Entropy-conjugacy + equivalence on Baire sets; what are the equivalence relations on homeomorphisms or maps on $S^1$ and subshifts?
I understand: Notions of similarity intermediate between topological conjugacy and measurable isomorphism a.e.. right?
Entropy-conjugacy refers to Bowen's paper .
See  for recent work.
See Problem 99
Various versions of ``conjugacy between topology and measure theory'' have been studied in the case of a mixing shift of finite type $S$ with a specified invariant Markov measure $\mu$. At one extreme, a conjugacy $\phi: (S,\mu ) \to (S,\mu')$ is only required to satisfy $\phi_* (\mu ) = \mu'$ with $\phi S = S'\phi$ on a set of full $\mu$ measure. At the other extreme, $\phi$ is also required to be a homeomorphism. Intermediate requirements on $\phi$ were studied by Keane and Smorodinsky, Parry, Schmidt, Marcus and Tuncel, Gomez ... For a brief review of this with references, and the still-open ``Good finitary conjecture'', see the discussion of Problem 9 in  (also available as ``OPSD'' at http://www.math.umd.edu/~mboyle/open/ ).
In another direction, one can consider homeomorphisms $S$ and $S'$ with respect to all invariant Borel probabilities (not just one), and ask for a topological or Borel conjugacy between the complements of sets of measure zero for all nonatomic invariant Borel probabilities. Hochman  proved striking universality results in this direction (for application of this work to $C^+$ diffeomorphisms, see ).
Finally, following Shelah and Weiss, one can consider $S$ and $S'$ as Borel systems, neglecting wandering sets; see  for dramatic progress on this long stalled study.