# Problem 35

## Primary tabs

Ergodic non hyperbolic automorphisms of $\mathbb{T}^n$. Are they quotients of subshifts of finite type? Do they satisfy specification?

## Tags

Tags:

### FL

Not really; cf [1]

### References

1. [lind1982dynamical] Lind DA.  1982.  Ergodic Theory and Dynamical Systems. 2:49–68.

### References

1. [marcus1980periodic] Marcus B.  1980.  Monats. fur Math.. 89

### Lemma 2.1 of this paper shows

Lemma 2.1 of this paper shows a specification-like property provided the gaps between specified blocks grow sub-linearly in the lengths of the orbit segments that are being constrained. (For ordinary specification, these gaps are of constant length).

### The ergodic nonhyperbolic

The ergodic nonhyperbolic toral automorphisms do not admit covers by shifts of finite type, but there are other symbolic representations. See [1].

### References

1. [LindenstraussSchmidt2005] Lindenstrauss E, Schmidt K.  2005.  Israel J. Math.. 149:227–266.

### The state of the art in

The state of the art in symbolic representations of toral automorphisms is contained in the paper *Representations of toral automorphisms* by Klaus Schmidt, Topology and its applications 205 (2016), 88-116 [1] .

### A complete description of the

A complete description of the specification properties of ergodic toral automorphisms is given in "Ergodic group automorphisms and specification", Douglas Lind, Springer Lecture Notes in Mathematics 729 (1978), 93-104. Toral automorphisms come in three flavors, depending on the behavior on the generalized eigenspace for eigenvalues of modulus one (if any), and the specification behavior is different for each flavor.