When do automorphisms of subshifts extend to automorphisms of the full shift?

Question

Let $A$ be a finite alphabet, $X$ = $(A^\mathbb{Z}, \sigma)$ the full shift, and $Y \subset X$ a subshift.

Question:

Are there any general results characterizing whether automorphisms of $(Y, \sigma)$ are likely to extend to automorphisms of $X$ (that is, whether automorphisms of $(Y, \sigma)$ are likely to be restrictions of automorphisms of $X$).

I suspect that the answer is that automorphisms generally do not extend. For example, it may be the case that for most subshifts (for some appropriate meaning of "most"), the only automorphisms that extend are powers of $\sigma$.

Any thoughts or references would be greatly appreciated!

Answer 1

If $\phi$ is an automorphism of $X$ and $Y$ is the set of points in $X$ of exact period $n$, then $\phi|_Y$ is an automorphism of $Y$. There is a subtle relationship between the sign of the permutation that $\phi$ induces on orbits of lengths $n$ for various $n$, and the value of a transfer function called the nth gyration number of $\phi$, called the sign-gyration compatibility condition, introduced by Boyle and Krieger [Periodic points and automorphisms of the shift, Trans. AMS 302 (1987), 125-149]. Using this, Kim, Roush, and Wagoner [Automorphisms of the dimension group and gyration numbers, Journ. AMS 5 (1992), 191-212] showed that there is a shift of finite type and a permutation of its fixed points that is not the restriction of an automorphism of the shift.

These ideas played a key role in the discovery by Kim, Roush, and Wagoner of counterexamples to what was then the main open problem in symbolic dynamics, namely the Shift Equivalence Conjecture. This conjecture was that a relatively easily computed invariant of shifts of finite type, the dimension triple, was a complete invariant for topological conjugacy.