Computability of topological entropy: From general systems to transformations on Cantor sets and the interval

Abstract

The dynamics of symbolic systems, such as multidimensional subshifts of finite type or cellular automata, are known to be closely related to computability theory. In particular, the appropriate tools to describe and classify topological entropy for this kind of systems turned out to be algorithmic. Part of the great importance of these symbolic systems relies on the role they have played in understanding more general systems over non-symbolic spaces. The aim of this article is to investigate topological entropy from a computability point of view in this more general, not necessarily symbolic setting. In analogy to effective subshifts, we consider computable maps over effective compact sets in general metric spaces, and study the computability properties of their topological entropies. We show that even in this general setting, the entropy is always a Σ 2 -computable number. We then study how various dynamical and analytical constrains affect this upper bound, and prove that it can be lowered in different ways depending on the constraint considered. In particular, we obtain that all Σ 2 -computable numbers can already be realized within the class of surjective computable maps over {0, 1} N , but that this bound decreases to Π 1 (or upper)-computable numbers when restricted to expansive maps. On the other hand, if we change the geometry of the ambient space from the symbolic {0, 1} N to the unit interval [0, 1], then we find a quite different situation -we show that the possible entropies of computable systems over [0, 1] are exactly the Σ 1 (or lower)-computable numbers and that this characterization switches down to precisely the computable numbers when we restrict the class of system to the quadratic family.

Hausdorff Dimension and Topological Entropies of a Solenoid

“…In 2002, Milnor [ 12 ] stated two questions related to the classical dynamical system: “Is entropy of it effectively computable?” “Given an explicit dynamical system and given , is it possible to compute the entropy with maximal error of ?” In most cases the answer is negative. For the recent results on computability of topological entropy, we recommend [ 13 , 14 ].…”

On entropies of block-gluing subshifts

“…A general point of view can be to specify the algorithmic complexity of the set of numbers realized as entropies depending on the dynamical properties of the subshift of finite type (see [5] for more details). An interesting dynamical property for subshift of finite type is the notion of c-block-gluing.…”

“…Now consider factors of length 5. By Proposition 11, one has that they are from the set 1 5 , 11011, 10101, 10001, 0 5 , 00100, 01010, 01110. Due to Proposition 5 for k = 5, we must prohibit at least 2 factors of length 5 -and we will now prove that we cannot prohibit more than one.…”

“…Indeed, consider a word u − * − v (i.e., u 1 u 2 u 3 u 4 * v 2 v 3 v 4 v 5 ): it has to be in L Z , so at least one of the words u and v must also be in L Z . So, without loss of generality, the prohibited words must be from 1 5 , 11011, 10101, 10001. We cannot prohibit 1 5 or 10001, since they are maximal (by Proposition 11); and we cannot prohibit at the same time 10101 and 11011 (consider 101 * 1011).…”

“…So, without loss of generality, the prohibited words must be from 1 5 , 11011, 10101, 10001. We cannot prohibit 1 5 or 10001, since they are maximal (by Proposition 11); and we cannot prohibit at the same time 10101 and 11011 (consider 101 * 1011). So, we can prohibit at most one factor of length 5, which gives the proof.…”