Entropy dimension of measure preserving systems

Abstract

The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight to analyze the entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate the joinings among entropy zero systems and prove the disjointness property among some classes of entropy zero systems using the dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.2010 Mathematics Subject Classification. Primary: 37A35, 37A05, 28D20.

Zero sequence entropy and entropy dimension

“…Particularly, it only works on topological systems with zero entropy. As an analogy of topological entropy dimension, the measure-theoretical version of the entropy dimension D µ (X, T ) for µ ∈ M (X, T ) was also defined to measure the growth rate of iterated partitions in [4]. It should be mentioned unfortunately that, there is no variational principle between the topological entropy dimension and the measure-theoretical entropy dimension (see [5, example 4.6]).…”

“…α) = sup S∈Eµ(T,α) D(S), E µ (T, α) = ∅ 0, E µ (T, α) = ∅ , D p µ (T, α) = sup S∈Pµ(T,α) D(S), P µ (T, α) = ∅ 0, P µ (T, α) = ∅ , D e µ (X, T ) = sup α∈P X D e µ (T, α),andD p µ (X, T ) = sup α∈P X D p µ (T, α).It is proved that D e µ (T, α) = D p µ (T, α) for any µ ∈ M (X, T ) and α ∈ P X (see[4] for details). Hence we define D µ (T, α) = D e µ (T, α) = D p µ (T, α) which is called the upper entropy dimension of α, and D µ (T, α) = D e µ (T, α) which is called the lower entropy dimension of α.…”

Relative entropy dimension of topological dynamical systems

“…systems are disjoint from minimal and entropy zero systems. Relevant results for measure-theoretic settings can be found in recent work by D. Dou, W. Huang and K. K. Park in [4]. In this paper we would like to introduce the notion of relative dimension tuples and the relative dimension set and prove that two extensions of disjoint relative dimension sets for all orders are disjoint over the same dynamic system under some conditions.…”

Relative entropy dimensions for amenable group actions

“…Although systems with positive entropy are much more complicated than those with zero entropy, zero entropy systems own various levels of complexity, and recently have been discussed in [3,4,6,7,9,13,18,22]. Those authors adopted various methods to classify zero entropy dynamical systems.…”