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.

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We show that a certain type of quasi finite, conservative, ergodic , measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic , measure preserving transformation which is not quasi finite; and consider distribution asymptotics of information showing that e.g. for Boole's transformation, information is asymptotically mod-normal with normalization ∝ √ n. Lastly we see that certain ergodic, probability preserving transformations with zero entropy have analogous properties and consequently entropy dimension of at most 1 2 .1991 Mathematics Subject Classification. 37A40, 60F05).

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.

Abstract. If an ergodic system has positive entropy, then the ShannonMcMillan-Breiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has OrnsteinWeiss' first return time property, which offers a method of computing the entropy via an orbit. We consider irrational rotations which are the simplest model of zero entropy. We prove that almost every irrational rotation has the analogous properties if properly normalized. However there are some irrational rotations that exhibit different behavior.