Problem 77

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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?

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FL

I understand: Notions of similarity intermediate between topological conjugacy and measurable isomorphism a.e.. right?

Entropy-conjugacy refers to

Entropy-conjugacy refers to Bowen's paper [1].

See [2] for recent work.

See Problem 99

References

1. [bowen1973topological] Bowen R.  1973.  Transactions of the American Mathematical Society. 184:125–136.
2. [boyle2014almost] Boyle M, Buzzi J.  2014.  arXiv preprint arXiv:1409.7330.

Various versions of 

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 [1] (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 [2] proved striking universality results in this direction (for application of this work to $C^+$ diffeomorphisms, see [3]).

Finally, following Shelah and Weiss, one can  consider $S$ and $S'$ as Borel systems, neglecting wandering sets; see [4] for dramatic progress on this long stalled study.

References

1. [Boyle2008open] Boyle M.  2008.  Geometric and probabilistic structures in dynamics. 469:69–118.
2. [hochman2013full] Hochman M.  2013.  Acta Appl. Math.. 126:187–201.