Entropy dimension of topological dynamical systems

Abstract

Abstract. We introduce the notion of topological entropy dimension to measure the complexity of entropy zero systems. It measures the superpolynomial, but subexponential, growth rate of orbits. We also introduce the dimension set, D(X, T ) ⊂ [0, 1], of a topological dynamical system to study the complexity of its factors. We construct a minimal example whose dimension set consists of one number. This implies the property that every nontrivial open cover has the same entropy dimension. This notion for zero entropy systems corresponds to the K-mixing property in measurable dynamics and to the uniformly positive entropy in topological dynamics for positive entropy systems. Using the entropy dimension, we are able to discuss the disjointness between the entropy zero systems. Properties of entropy generating sequences and their dimensions have been investigated.

Topological Entropy Dimension and Directional Entropy Dimension for ℤ2-Subshifts

“…As for the case of directional entropy, for v ∈ Z 2 , D(v) coincides with the topological upper entropy dimension [14] of the Z-topological dynamical system (X, σ v ). One can see that D(v) = D(tv) for all t > 0.…”

“…For details on symbolic dynamics, see [16], and, for topological entropy dimension of Z-actions, see [14].…”

“…In this section, we present equivalent formulations for two-dimensional entropy dimension using the entropy generating shape, which generalizes the notion of entropy generating sequence for one-dimensional case in [14]. The argument directly extends to the case of Z d -actions for any integer d > 2.…”

“…Clearly, D e ≤ D * e . In ( [14], Theorems 3.8 and 3.10), it was shown that if X is a Z-subshift, then D(X) equals the supremum of D(S) over all entropy generating sequences S for X. One may check that the proof is valid for Z 2 -subshifts with a little modification.…”

“…Topological entropy dimension has been introduced and studied in [14,15] to classify the growth rate of the orbits of zero entropy systems. For example, any positive entropy Z 2 -subshift has the orbit growth rate in the order of 2 n 2 , while the three dot model has the orbit growth rate in the order of 2 n .…”