It is well known that any continuous piecewise monotone interval map f with positive topological entropy $h_{top}\left(f\right)$ is semiconjugate to some piecewise affine map with constant slope $e^{h_{top}\left(f\right)}$. We prove this result for a class of Markov countably piecewise monotone continuous interval maps.

We associate a $C^{*}$-algebra to a locally compact Hausdorff groupoid with the property that the range map is locally injective. The construction generalizes J. Renault’s reduced groupoid $C^{*}$-algebra of an étale groupoid and has the advantage that it works for the groupoid arising from a locally injective dynamical system by the method introduced in increasing generality by Renault, Deaconu and Anantharaman-Delaroche. We study the $C^{*}$-algebras of such groupoids and give necessary and sufficient conditions...

We study the action of G = SL(2,ℝ), viewed as a group definable in the structure M = (ℝ,+,×), on its type space $S_G\left(M\right)$. We identify a minimal closed G-flow I and an idempotent r ∈ I (with respect to the Ellis semigroup structure * on $S_G\left(M\right)$). We also show that the “Ellis group” (r*I,*) is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.

We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $𝕋$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $𝕋$, and Bohr if it is recurrent for all products of rotations on $𝕋$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which...

Let (G,X) be a transformation group, where X is a locally compact Hausdorff space and G is a compact group. We investigate the stable rank and the real rank of the transformation group C*-algebra C₀(X)⋊ G. Explicit formulae are given in the case where X and G are second countable and X is locally of finite G-orbit type. As a consequence, we calculate the ranks of the group C*-algebra C*(ℝⁿ ⋊ G), where G is a connected closed subgroup of SO(n) acting on ℝⁿ by rotation.