Inspired by a twist map theorem of Mather. we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map g to the k-fold cover. For each irrational in the rotation set’s interior, the collection of the k-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$-dimensional ball with the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool used is a generalization of a construction of Hedlund and Morse that generates symbolic analogs of these k-fold well-ordered invariant sets.

We consider skew-product maps over circle rotations $x\mapsto x+\alpha \;(\mod 1)$ with factors that take values in ${\textrm {SL}}(2,{\mathbb {R}})$ . In numerical experiments, with $\alpha $ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.

In this paper, we show that each element in the convex hull of the rotation set of a compact invariant chain transitive set is realized by a Birkhoff solution, which is an improvement of the fundamental lemma of T. Zhou and W.-X. Qin [Pseudo solutions, rotation sets, and shadowing rotations for monotone recurrence relations. Math. Z. 297 (2021), 1673–1692] in the study of rotation sets for monotone recurrence relations. We then investigate the properties of rotation sets assuming the system has zero topological entropy. The rotation set for a Birkhoff recurrence class is a singleton and the forward and backward rotation numbers are identical for each solution in the same Birkhoff recurrence class. We also show the continuity of rotation numbers on the set of non-wandering points. If the rotation set is upper-stable, then we show that each boundary point is a rational number, and we also obtain a result of bounded deviation.

We prove that, if f is a homeomorphism of the 2-torus isotopic to the identity whose rotation set is a non-degenerate segment and f has a periodic point, then it has uniformly bounded deviations in the direction perpendicular to the segment.

Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

We provide a complete characterization of periodic point free homeomorphisms of the $2$ -torus admitting irrational circle rotations as topological factors. Given a homeomorphism of the $2$ -torus without periodic points and exhibiting uniformly bounded rotational deviations with respect to a rational direction, we show that annularity and the geometry of its non-wandering set are the only possible obstructions for the existence of an irrational circle rotation as topological factor. Through a very precise study of the dynamics of the induced $\rho $ -centralized skew-product, we extend and generalize considerably previous results of Jäger.

We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$ , $d\ge 2$ . In the conservative setting, we prove that there exists a Baire residual subset of the set $\text {Homeo}_{0, \lambda }(\mathbb T^2)$ of conservative homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb T^2$ , and that it carries full topological pressure and full metric mean dimension. Moreover, we prove that for every $d\ge 2$ the rotation set of $C^0$ -generic conservative homeomorphisms on $\mathbb T^d$ is convex. Related results are obtained in the case of dissipative homeomorphisms on tori. The previous results rely on the description of the topological complexity of the set of points with wild historic behavior and on the denseness of periodic measures for continuous maps with the gluing orbit property.

Let $\unicode[STIX]{x1D6E4}$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\unicode[STIX]{x1D6E4}$ on the circle is either trivial or semiconjugate to a unique minimal action on the so-called simple circle.

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ isotopic to the identity. A set of hypotheses is presented, called a fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincaré disk $\mathbb{D}$), denoted by $\widetilde{f},$ has complicated and rich dynamics. In this context, we generalize results that hold for homeomorphisms of the torus isotopic to the identity when their rotation sets contain zero in the interior. In particular, for $C^{1+\unicode[STIX]{x1D716}}$ diffeomorphisms, we show the existence of rotational horseshoes having non-trivial displacements in every homotopical direction. As a consequence, we found that the homological rotation set of such an $f$ is a compact convex subset of $\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are realized by compact $f$-invariant sets and by periodic orbits in the rational case. Also, $f$ has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. This implies, in case where $f$ is area preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.

An isotopic to the identity map of the 2-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the 2-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular, we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the 2-torus has a global fixed point.

We show that if there exists a counter example for the rational case of the Franks–Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set.

In this paper we consider $C^{\infty }$-generic families of area-preserving diffeomorphisms of the torus homotopic to the identity and their rotation sets. Let $f_{t}:\text{T}^{2}\rightarrow \text{T}^{2}$ be such a family, $\widetilde{f}_{t}:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ be a fixed family of lifts and $\unicode[STIX]{x1D70C}(\widetilde{f}_{t})$ be their rotation sets, which we assume to have interior for $t$ in a certain open interval $I$. We also assume that some rational point $(p/q,l/q)\in \unicode[STIX]{x2202}\unicode[STIX]{x1D70C}(\widetilde{f}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$, and we want to understand the consequences of the following hypothesis: for all $t>\overline{t}$, $t\in I$, $(p/q,l/q)\in \text{int}(\unicode[STIX]{x1D70C}(\widetilde{f}_{t}))$. Under these very natural assumptions, we prove that there exists a $f_{\overline{t}}^{q}$-fixed hyperbolic saddle $P_{\overline{t}}$ such that its rotation vector is $(p/q,l/q)$. We also prove that there exists a sequence $t_{i}>\overline{t}$, $t_{i}\rightarrow \overline{t}$, such that if $P_{t}$ is the continuation of $P_{\overline{t}}$ with the parameter, then $W^{u}(\widetilde{P}_{t_{i}})$ (the unstable manifold) has quadratic tangencies with $W^{s}(\widetilde{P}_{t_{i}})+(c,d)$ (the stable manifold translated by $(c,d)$), where $\widetilde{P}_{t_{i}}$ is any lift of $P_{t_{i}}$ to the plane. In other words, $\widetilde{P}_{t_{i}}$ is a fixed point for $(\widetilde{f}_{t_{i}})^{q}-(p,l)$, and $(c,d)\neq (0,0)$ are certain integer vectors such that $W^{u}(\widetilde{P}_{\overline{t}})$ do not intersect $W^{s}(\widetilde{P}_{\overline{t}})+(c,d)$, and these tangencies become transverse as $t$ increases. We also prove that, for $t>\overline{t}$, $W^{u}(\widetilde{P}_{t})$ has transverse intersections with $W^{s}(\widetilde{P}_{t})+(a,b)$, for all integer vectors $(a,b)$, and thus one may consider that the tangencies above are associated to the birth of the heteroclinic intersections in the plane that do not exist for $t\leq \overline{t}$.

A classical article by Misiurewicz and Ziemian (J. Lond. Math. Soc.40(2) (1989), 490–506) classifies the elements in Homeo$_{0}(\mathbf{T}^{2})$ by their rotation set $\unicode[STIX]{x1D70C}$, according to wether $\unicode[STIX]{x1D70C}$ is a point, a segment or a set with nonempty interior. A recent classification of nonwandering elements in Homeo$_{0}(\mathbf{T}^{2})$ by Koropecki and Tal was given in (Invent. Math.196 (2014), 339–381), according to the intrinsic underlying ambient space where the dynamics takes place: planar, annular and strictly toral maps. We study the link between these two classifications, showing that, even abroad the nonwandering setting, annular maps are characterized by rotation sets which are rational segments. Also, we obtain information on the sublinear diffusion of orbits in the—not very well understood—case that $\unicode[STIX]{x1D70C}$ has nonempty interior.

Nous montrons des résultats d’existence de points fixes communs pour des homéomorphismes du plan ${ \mathbb{R} }^{2} $ ou la sphère ${ \mathbb{S} }^{2} $, qui commutent deux à deux et préservent une mesure de probabilité. Par exemple, nous montrons que des ${C}^{1} $-difféomorphismes ${f}_{1} , \ldots , {f}_{n} $ de ${ \mathbb{S} }^{2} $ suffisamment proches de l’identité, qui commutent deux à deux, et qui préservent une mesure de probabilité dont le support n’est pas réduit à un point, ont au moins deux points fixes communs.

Using reparametrizations of linear flows, we show that there exist area-preserving real analytic maps of the three-dimensional torus that are ‘mixing of all orders’ and do not enjoy the monotone shrinking target property. Prior to that, we give a short proof of a result of Kurzweil from 1955: namely, that a translation $T_\alpha$ of the torus $\mathbb{T}^d$ has the monotone shrinking target property if and only if the vector $\alpha$ is badly approximable (that is, of constant type).