This note is a companion paper to ‘Monodromie et classification topologique des germes de feuilletages holomorphes’ (in French) by Marin and Mattei. In their paper the authors develop a very complete theory about topologically conjugate singularities of foliations. However, their techniques require the topological conjugacies to be transversely holomorphic. The purpose of the present note is to show that this assumption can be made without restricting the generality of their results.

At the end of the last century Vershik introduced some dynamical systems, called polymorphisms. Systems of this kind are multivalued self-maps of an interval, where (roughly speaking) each branch has some probability. By definition, the standard Lebesgue measure should be invariant. Unexpectedly, some class of polymorphisms appeared in the problem of destruction of an adiabatic invariant after a multiple passage through a separatrix. We discuss ergodic properties of polymorphisms from this class.

We use the results of Neshveyev and Størmer to show that for a generic shift on a C*-algebra associated with a bitstream the Voiculescu topological entropy is strictly larger that the supremum of topological entropies of its classical subsystems.

We study the dynamic properties of rank-one ℤd actions as a function of the geometry of the shapes of the towers generating the action. Some basic properties require only minimal restrictions on the geometry of the towers. Our main results concern the directional entropy of rank-one ℤd actions with rectangular tower shapes, where we show that the geometry of the rectangles plays a significant role. We show that for each n≤d there is an n-dimensional direction with entropy zero. We also show that if the growth in eccentricity of the rectangular towers is sub-exponential, then all directional entropies are zero. An example of D. Rudolph shows that, without a restriction on eccentricity, a positive entropy direction is possible.

Let X be a metrizable space with a continuous group or semi-group action G. Let D be a non-empty subset of X. Our problem is how to choose a fixed number of sets in {g−1D; g∈G}, say σ−1D with σ∈τ, to maximize the cardinality of the partition ℙ({σ−1D; σ∈τ}) generated by them. Let An infinite subset Σ of G is called an optimal position of the triple (X,G,D) if holds for any k=1,2,… and τ⊂Σ with #τ=k. In this paper, we discuss examples of the triple (X,G,D) admitting or not admitting an optimal position. Let X=G=ℝn  (n≥1) , where the action g∈G to x∈X is the translation x−g. If D is the n-dimensional unit ball, then holds and the triple (X,G,D) admits an optimal position. In fact, if n≥2 and Σ is an infinite subset of G such that for some δ with 0<δ<1 , Σ⊂{x∈ℝn ; ‖x‖=δ}, and that any subset of Σ with cardinality n+1 is not on a hyperplane, then Σ is an optimal position of the triple (X,G,D) . We have determined the primitive factor of the uniform sets coming from these optimal positions. We also show that in the above setting with n=2 and the unit square D′ in place of the unit disk D, the maximal pattern complexity is unchanged and p*X,G,D′(k)=k2 −k+2 , but there is no optimal position.

This paper deals with non-singular Smale flows on oriented 3-manifolds. We shall show a relation between the properties of invariant sets of a Smale flow and a kind of Whitehead torsion of the underlying manifold.