The Jewett-Krieger Construction for Tilings

Theorem 1 (Jewett-Krieger). Let (X,A , μ, S) be an ergodic dynamical system. Then there is a compact space Y and a homeomorphism T on Y , which is uniquely ergodic and minimal, such that if ν denotes the unique T -invariant probability measure on Y and B the Borel σalgebra of Y , the ergodic dynamical system (Y,B, ν, T ) is equivalent to (X,A , μ, S). In addition, (Y,B, ν, T ) can be constructed in a such a way that its topological entropy is as close as wished from the Kolmogorov-Sinai entropy of (X,A , μ, S).