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In this paper, we completely characterize locally compact flows G G of homeomorphisms of connected manifolds M M by proving that they are either circle groups or real groups. For M = R m M = \mathbb R^m , we prove that every recurrent element in G G is periodic, and we obtain a generalization of the result of Yang [Hilbert’s fifth problem and related problems on transformation groups, American Mathematical Society, Providence, RI, 1976, pp. 142–146.] by proving that there is no nontrivial locally compact flow on R m \mathbb R^m in which all elements are recurrent.

Let $G = \lbrace h_t \; \vert \; t \in \mathbb{R} \rbrace$ be a continuous flow on a connected $n$-manifold $M$. The flow $G$ is said to be strongly reversible by an involution $\tau$ if $h_{-t} = \tau h_t \tau$ for all $t \in \mathbb{R}$, and it is said to be periodic if $h_s = $ identity for some $s \in \mathbb{R}^\ast$. A closed subset $K$ of $M$ is called a global section for $G$ if every orbit $G(x)$ intersects $K$ in exactly one point. In this paper, we study how the two properties “strongly reversible” and “has a global section” are related. In particular, we show that if $G$ is periodic and strongly reversible by a reflection, then $G$ has a global section.

Let $$\text {PL}(S)$$ be the group of piecewise linear homeomorphisms of the circle S which are locally affine at all but a finite number of points of S (called break points) and $$\text {PL}^+(S)$$ be the subgroup of $$\text {PL}(S)$$ whose elements are order preserving. An element in $$\text {PL}^+(S)$$ is called reversible in $$\text {PL}^+(S)$$ if it is conjugate to its inverse in $$\text {PL}^+(S)$$ . In this paper, we answer partially one of O’Farrell and Short’s problems in their recent book (O’Farrell and Short in Reversibility in dynamics and group theory. London Mathematical Society Lecture Note Series: 416. Cambridge University Press, Cambridge, 2014). We characterize the reversible elements in $$\text {PL}^+(S)$$ and also perform a similar characterization in the full group $$\text {PL}(S)$$ . In particular, we give a complete characterization of reversibility in $$\text {PL}^+(S)$$ and $$\text {PL}(S)$$ of elements f of $$\text {PL}^+(S)$$ having the (D)-property (i.e. for which the product of the f-jumps along the full orbit of any break point is one).

We consider abelian nonautonomous dynamical systems F = ( f n ) on compact spaces X which are moreover transitive; that is, the limit set of some point x ∈ X coincides with the whole space X, and we extend some results of [1] . More precisely, we show that such F must be either pointwise sensitive or uniformly rigid and almost equicontinuous. Moreover, we prove that if the set of equicontinuity points of F has nonempty interior, then it coincides with the set of transitive points and F will be equicontinuous if and only if it is minimal. Next, we assume that the sequence ( f n ) converges uniformly to a transitive map f, and we prove that if f is minimal or equicontinuous then F is either sensitive or equicontinuous, minimal and uniformly rigid. We also study chaos for F, and we prove that Devaney chaos implies Li-Yorke chaos.

This paper is the paper announced in [Be2, References [2]]. We show that every compact abelian group of homeomorphisms of $\mathbb{R}^3$ is either zero-dimensional or equivalent to a subgroup of the orthogonal group O(3). We prove a similar result if we replace $\mathbb{R}^3$ by $\mathbb{S}^3$, and we study regular homeomorphisms that are conjugate to their inverses.

Let G be a positively equicontinuous flow of homeomorphisms of a locally compact metric space E. We show how the dynamics of such a flow are rich. We study when regularly almost periodic elements in G are periodic and we describe orbits and their limit sets. In particular, we show that the limit set L(G ) of G is a closed subset on which G is equicontinuous and that if G moreover has closed orbits, then any w-limit set is a periodic orbit.

We establish some equivalent conditions to the Hilbert–Smith conjecture on connected n-manifolds M. For n = 3; we show that regularly almost periodic homeomorphisms of M are periodic; this result extends Theorem 5.34 of Gottschalk and Hedlund (Topological Dynamics. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, 1956). For the special case of \({M = \mathbb{R}^3}\), we extend the result of Brechner (Pac J Math 59(2):367–374, 1975) saying that “almost periodic homeomorphisms of the plane are periodic” to \({\mathbb{R}^3}\), and we show that any compact abelian group of homeomorphisms of \({\mathbb{R}^3}\) is either finite or topologically equivalent to a subgroup of the orthogonal group O(3).

In this paper we study recurrent and almost periodic homeomorphisms on the Euclidean space R m ; we give conditions under which recurrent implies periodic. On the other hand we give properties of elements of compact groups of diffeomorphisms on manifolds.