Abstract
Construction sequences are a general method of building symbolic shifts that capture cut-and-stack constructions and are general enough to give symbolic representations of Anosov-Katok diffeomorphisms. We show here that any finite entropy system that has an odometer factor can be represented as the limit of a special class of construction sequences, the odometer based construction sequences. These naturally correspond to those cut-and-stack constructions that do not use spacers. The odometer based construction sequences can be constructed to have the small word property and every Choquet simplex can be realized as the simplex of invariant measures of the limit of an odometer based construction sequence.
References
T. Adams, S. Ferenczi and K. Petersen, Constructive symbolic presentations of rank one measure-preserving systems, Colloquium Mathematicum 150 (2017), 243–255.
F. Beleznay and M. Foreman, The complexity of the collection of measure-distal transformations, Ergodic Theory and Dynamical Systems 16 (1996), 929–962.
T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel Journal of Mathematics 74 (1991), 241–256.
T. Downarowicz and Y. Lacroix, Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows, Studia Mathematica 130 (1998), 149–170.
S. Ferenczi, Systems of finite rank, Colloquium Mathematicum 73 (1997), 35–65.
M. Foreman, D. J. Rudolph and B. Weiss, The conjugacy problem in ergodic theory, Annals of Mathematics 173 (2011), 1529–1586.
M. Foreman and B. Weiss, From odometers to circular systems: a global structure theorem, Journal of Modern Dynamics 15 (2019), 345–423.
M. Foreman and B. Weiss, A symbolic representation for Anosov-Katok systems, Journal d’Analyse Mathématique 137 (2019), 603–661.
M. Foreman and B. Weiss, Measure preserving diffeomorphisms are unclassifiable, Journal of the European Mathematical Society, to appear.
W. Krieger, On entropy and generators of measure-preserving transformations, Transactions of the American Mathematical Society 149 (1970), 453–464.
B. Weiss, Strictly ergodic models for dynamical systems, Bulletin of the American Mathematical Society 13 (1985), 143–146.
B. Weiss, Single Orbit Dynamics, CBMS Regional Conference Series in Mathematics, Vol. 95, American Mathematical Society, Providence, RI, 2000.
S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 67 (1984), 95–107.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author would like to acknowledge partial support from NSF grant DMS-2100367.
Rights and permissions
About this article
Cite this article
Foreman, M., Weiss, B. Odometer Based Systems. Isr. J. Math. 251, 327–364 (2022). https://doi.org/10.1007/s11856-022-2439-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-022-2439-7