MathDBLindenstrauss's small boundary property conjecture
Lindenstrauss's small boundary property conjecture
A free dynamical system is a topological dynamical system in which for every and every nonzero integer . Its mean dimension is the dynamical dimension invariant associated with , and the small boundary property is the corresponding topological regularity property for which the boundary of every sufficiently refined open set has vanishing dynamical dimension. Lindenstrauss's conjecture. If a free dynamical system has zero mean dimension, then it has the small boundary property. The source identifies this as equivalent to the preceding marker-property conjecture and attributes it to Lindenstrauss; it is presented as an unresolved conjecture in the discussion of marker properties.
Equivalent formulations 1
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
Lindenstrauss's small boundary property conjecture
Let be a dynamical system. It has the small boundary property if every point has a neighborhood whose boundary is a null set. The system is aperiodic if it has no periodic points, and it has zero mean dimension if its mean dimension is zero.
Lindenstrauss's conjecture. Every aperiodic dynamical system with zero mean dimension has the small boundary property.
The source states that this conjecture is equivalent to the zero mean dimension conjecture for the marker property. It is still open in general.
source: Ruxi Shi, “Finite mean dimesnion and marker property”, arXiv:2102.12197 (2021).
Sources & referencesView supporting material
Primary source
Masaki Tsukamoto, Mitsunobu Tsutaya and Masahiko Yoshinaga, “G-index, topological dynamics and marker property”, arXiv:2012.15372 (2020).
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