Toeplitz minimal flows which are not uniquely ergodic

Toeplitz subshift whose automorphism group is not finitely generated

“…Our methods are very elementary, and we do not need much of the theory of Toeplitz subshifts -in particular, while the discussion below is strongly related to the maximal equicontinuous factor of a Toeplitz subshift, we will not discuss this factor explicitly, but only its finite factors. 2 The lemmas that follow can be extracted from, or found directly in, any reference that discusses the maximal equicontinuous factor of a Toeplitz subshift [16,23,9], but we give a self-contained presentation with a more combinatorial point of view.…”

0-1 sequences of the Thue-Morse type and Sarnak’s conjecture

“…This factor is determined by so called Thue-Toeplitz sequence 4 . In order to complete the proof, we show that Sarnak's conjecture is true for all dynamical systems given by so called regular Toeplitz sequences [12], [26], see Section 6. This general result, in Section 7, enables us to conclude the validity of Sarnak's conjecture for all dynamical systems arising from the Thue-Morse type 0 − 1-sequences.…”

“…. It can be proved [12], [26] that then there is an increasing sequence (p n ), p n |p n+1 such that for each n ≥ 1, z can be represented as z = C n C n . .…”

“…The dynamical system generated by a regular Toeplitz sequence is uniquely ergodic and has zero entropy [12], [26] (see also [6]). Moreover, dynamical systems generated by Toeplitz sequences are so called almost 1-1 extensions of their maximal equicontinuous factors which are (p n )-odometers.…”

Rotation, entropy, and equilibrium states

“…A convenient class of subshifts for realising this are Toeplitz subshifts (see for example [58]). These subshifts are always minimal, but can be very far from uniquely ergodic.…”