Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Abstract

A Borel Zd dynamical system (X,S) is ‘almost Borel universal’ if any free Borel Zd dynamical system (Y,T) of strictly lower entropy is isomorphic to a Borel subsystem of (X,S), after removing a null set. We obtain and exploit a new sufficient condition for a topological Zd dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a ‘generic’ homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non‐uniform specification implies almost Borel universality, and that 3‐colorings in Zd and dimers in Z2 are almost Borel universal.

Some Remarks on the Notion of Bohr Chaos and Invariant Measures

“…In section 4 we prove that invertible topological dynamical systems admitting the specification property are Bohr chaotic. Such systems are known to be universal for probability measure preserving systems with entropy less than theirs (see [1]). It is not known to us whether such a universality is by itself a sufficient condition for a topological dynamical system to be Bohr chaotic.…”

“…T i Y, T is a topological dynamical system that fulfills the requirement of the remark following the proof of theorem 4.2.1, thus it is Bohr chaotic and thus so is (X, T ). By lemma 4.2.2, there exist two different orbit segments x (1) , T x (1) , . .…”

“…It is left to find words α and β for which the parsing will indeed be unique. Taking α and β to be two words as in lemma 4.1.2 for r = 1 3 , double parsing implies l − l 3 − u < l 3 which is equivalent to l < 3u. So taking l ≥ 3u guarantees that the parsing will be unique.…”

“…T i Y -that have an ε 0 -overlap and ε 0 -self-overlaps all smaller than l ′ 4 and l ′ ≥ 3u δ , where u δ is the the specification spacing of Y, T k for δ. By appending at most k − 1 orbit coordinates to the left and to the right we can obtain two different orbit segments of length l satsifying l ′ ≤ l ≤ l ′ + 2k − 2: y (1) , T y (1) , . .…”

“…. , T (l−1)k y (2) -where y (1) , y (2) ∈ Y -that have an ε 0 -overlap and ε 0 -self-overlaps all smaller than l 3 and l ≥ 3u δ .…”