Universal minimal topological dynamical systems

Rokhlin (1963) showed that any aperiodic dynamical system with finite entropy admits a countable generating partition. Krieger (1970) showed that aperiodic ergodic systems with entropy < log a, admit a generating partition with no more than a sets. In Symbolic Dynamics terminology, these results can be phrased- ℕ^ℤ is a universal system in the category of aperiodic systems, and [a]^ℤ is a universal system for aperiodic ergodic systems with entropy < log a. Weiss ([We89], 1989) presented a Minimal system, on a Compact space (a subshift of [InlineMediaObject not available: see fulltext.]) which is universal for aperiodic systems. In this work we present a joint generalization of both results: given ε, there exists a minimal subshift of [a]^ℤ, universal for aperiodic ergodic systems with entropy < log a - ε.