Problem 146

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  1. Subshifts of finite type have good quotients with fixed points.
  2. Given a periodic point $p$  in $\Omega _i$, is there a Markov partition with $p$ in the interior of a rectangle?
  3. If $\partial^s \mathcal{C} \ne \emptyset$ does $\partial^s \mathcal{C}$ contain a periodic point?
  4. Do two subshifts of finite type with the same entropy have a common good quotient?
  5. $\Sigma _A, \Sigma _B$ aperiodic, $h (\sigma _A ) < h(\sigma_B). $ Does $\sigma_A |\Sigma_A$ embed in $\sigma_B |\Sigma_B$?


In parts a and d, "good quotient" probably means something like the quotient dynamical system is expansive or hyperbolic and the quotient mapping is finite-to-one or 1-1 a.e. (w.r.t. any fully supported ergodic measure). In both a and d, it is too much to expect an SFT quotient; how about a sofic quotient? See [1] and [2].

In part c,  $\mathcal{C}$ is a Markov partition for an Axiom A basic set; the boundary of each element of $ \mathcal{C}$ is the union of a stable boundary and unstable boundary, and the union of all stable boundaries is denoted $\partial^s \mathcal{C}$.

In part e, by aperiodic he means mixing. Assuming a condition on the numbers of periodic points, Krieger [3] proved existence of such an imbedding. See also [4].


Kitchens (a Bowen grandson) showed there are mixing shifts of finite type with no equal entropy SFT factor with a fixed point. See e.g. Example 4.2.12 of Kitchens' book [1]

Every shift of finite type has an equal entropy sofic quotient with a fixed point. (Form the quotient system by collapsing a single finite orbit to a point, and identifying no other points.)

I don't know whether every mixing SFT has an equal entropy quotient with canonical coordinates (local product structure) and a fixed point. 

b. A periodic point $p$ in the interior has a unique preimage in the covering SFT.  An answer ``no'' to this question would reveal  interesting nonhomogenity of the
map on the basic set, and the zeta function counting such $p$ (if they exist) would be an interesting invariant. 

c. The paper [2] should be relevant to this question. 

d. By work of Nasu or Kitchens, there are equal entropy mixing shifts of finite type with no common equal entropy SFT factor, as argued by Kitchens. See e.g. Example 4.2.12 of Kitchens' book [1]. (In contrast, by the Adler-Marcus theorem, two equal entropy mixing shifts of finite type have a common mixing SFT extension by 1-1 a.e. factor maps.)

Regarding other ``good quotient'' possibilities: must two equal entropy mixing SFTs  have a common equal entropy factor which is sofic? expansive? expansive with canonical coordinates? Must there be any common equal entropy factor?  For the sofic question, ideas around Nasu's core matrix might be relevant, as in the paper [3] giving an example of an entropy log n sofic shift which (in contrast to the SFT case) does not factor onto the full shift on n symbols. 


  1. [Kitchens1998book] Kitchens BP.  1998.  Symbolic dynamics. Universitext. :x+252.
  2. [AshleyKitchensStafford1992] Ashley J, Kitchens B, Stafford M.  1992.  Boundaries of {M}arkov partitions. Trans. Amer. Math. Soc.. 333:177–201.
  3. [TrowWilliams1996] Trow P, Williams S.  1993.  Core dimension group constraints for factors of sofic shifts. Ergodic Theory Dynam. Systems. 13:213–224.

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