Problem 35

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



Not really; cf [1]


  1. [lind1982dynamical] Lind DA.  1982.  Dynamical properties of quasihyperbolic toral automorphisms. Ergodic Theory and Dynamical Systems. 2:49–68.

See also [1].


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


Douglas Lind's picture

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] .


Douglas Lind's picture

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. 

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