Topological Dimension and Dynamical Systems

Odderon and pomeron as fractal dimension in pp and p̄p total cross-sections

“…This definition coincides with the usual notion of dimension for manifolds and of course, it assumes only integer values. See, for instance, Coornaert and references therein for a more detailed study [37].…”

Minimal subshifts of arbitrary mean topological dimension

“…Let y be a limit point of the sequence (g −1 n x) n≥1 . It is easy to see that S does not occur in y, which contradicts to (2).…”

“…When the countable infinite amenable group, denoted by G, has subgroups of arbitrarily large finite index, given a polyhedron P and a nonnegative real number ρ which is no more than the topological dimension of P , Coornaert and Krieger [3] (see also [2]) constructed a closed subshift X ⊂ P G having mean topological dimension ρ. Hence in this situation, the mean topological dimension of G-systems can take all values in [0, +∞].…”

Karol Borsuk 1905–1982

“…Set n ∈ N. Let m = n|K|. We now follow a proof of Proposition 9.2.13 from [8] to show that there exists t 0 ∈ G such that the set F m t −1 0 ∩ K = {k ∈ K : kt 0 ∈ F m } is an n-Følner for K. Let T ⊂ G be a complete set of representatives of the right cosets of K in G. Clearly, every g ∈ G can be uniquely written in the form g = ht with h ∈ K and t ∈ T . We then have:…”