In this paper we study some conjugacy invariants (moduli) for discrete two dimensional dynamical systems, with a homoclinic tangency. We show that the modulus obtained by Palis in the heteroclinic case also turns up in the case considered here. We also present two new conjugacy invariants.

Let f be a continuous map of the circle into itself of degree one. We introduce the notion of rotation algorithms. One of these algorithms associates each z ∈ S1 with an interval, the so-called speed interval S(z, f), which is contained in the rotation interval ρ(f) of f. In contrast with the rotation set ρ(z, f), the interval S(z, f) sometimes allows us to ascertain that ρ(f) is non-degenerate, by using only finitely many elements of {fn (z) | n ≥ 0}. We further show that all choices for ρ(z, f) and S(z, f) occur, for certain z ∈ S1 provided that ρ(z, f) ⊂ S(z, f) ⊂ ρ(f).

In this paper, we study the dynamics of the family of rational maps with two parameters

$$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$

$n\geq 2$

$a,b\in \mathbb{C}^{\ast }$

$f_{a,b}$

Let α1,…,αn be irrational numbers which are linearly independent over the rationals, and f1,…,fn commuting germs of holomorphic diffeomorphisms in ℂ such that fk(0)=0,f′k(0)=e2πiαk,k=1,…,n. Moser showed that f 1,…,fn are simultaneously linearizable (i.e. conjugate by a germ of holomorphic diffeomorphism h to the rigid rotations Rαk(z)=e2πiαkz) if the vector of rotation numbers (α 1,…,αn) satisfies a Diophantine condition. Adapting Yoccoz’s renormalization to the setting of commuting germs, we show that simultaneous linearization holds in the presence of a weaker Brjuno-type condition ℬ(α1,…,αn)<+∞, where ℬ(α1,…,αn) is a multivariable analogue of the Brjuno function. If there are no periodic orbits for the action of the germs f1,…,fn in a neighbourhood of the origin, then a weaker arithmetic condition ℬ′ (α 1,…,αn)<+∞ analogous to Perez-Marco’s condition for linearization in the absence of periodic orbits is shown to suffice for linearizability. Normalizing the germs to be univalent on the unit disc, in both cases the Siegel discs are shown to contain discs of radii for some universal constant C.

In this paper we show how to apply classical probabilistic tools for partial sums $\sum _{j=0}^{n-1}\varphi \circ \tau ^j$ generated by a skew product $\tau $, built over a sufficiently well-mixing base map and a random expanding dynamical system. Under certain regularity assumptions on the observable $\varphi $, we obtain a central limit theorem (CLT) with rates, a functional CLT, an almost sure invariance principle (ASIP), a moderate-deviations principle, several exponential concentration inequalities and Rosenthal-type moment estimates for skew products with $\alpha $-, $\phi $- or $\psi $-mixing base maps and expanding-on-average random fiber maps. All of the results are new even in the uniformly expanding case. The main novelty here (in contrast to [2]) is that the random maps are not independent, they do not preserve the same measure and the observable $\varphi $ depends also on the base space. For stretched exponentially ${\alpha }$-mixing base maps our proofs are based on multiple correlation estimates, which make the classical method of cumulants applicable. For $\phi $- or $\psi $-mixing base maps, we obtain an ASIP and maximal and concentration inequalities by establishing an $L^\infty $ convergence of the iterates ${\mathcal K}^{\,n}$ of a certain transfer operator ${\mathcal K}$ with respect to a certain sub-${\sigma }$-algebra, which yields an appropriate (reverse) martingale-coboundary decomposition.

We show that among Cr extensions (r>0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group ℋn of dimension 2n+1, those that avoid an obvious obstruction to topological transitivity are generically topologically transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with a compact commutator subgroup (for example ℋn/ℤ), among those that avoid the obvious obstruction, topological transitivity is open and dense.

We obtain quenched almost sure invariance principles (with convergence rates) for random Young towers if the average measure of the tail of return times to the base of random towers decays sufficiently fast. We apply our results to some independent and identically distributed perturbations of some non-uniformly expanding maps. These imply that the random systems under study tend to a Brownian motion under various scalings.

We construct, for each irrational number \alpha, a minimal C^1-diffeomorphism of the circle with rotation number \alpha which is not ergodic with respect to the Lebesgue measure.

In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb {Z}^q$ -systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, directional discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb {Z}^q$ -system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb {Z}^q$ -system has directional discrete spectrum along q linearly independent directions if and only if it has discrete spectrum.

Let G be a solvable Lie group. Let $\chi$ be a character on G with values in $\mathbb R$. Let $\mathfrak B$ be a Banach space on which G is made to act by bounded operators, with the equality $\Vert gb\Vert=\chi(g)\Vert b\Vert$ satisfied for all $g\in G$ and $b\in\mathfrak B$. A criterion is given on the pair $(G,\chi)$ for the cohomology group $H^1(G,\mathfrak B)$ to vanish. Using this criterion, we classify up to Cr conjugacy all the volume-preserving locally free Cr actions of G on closed manifolds with dimension $\dim(G)+1$, for G belonging to a wide class of solvable Lie groups.

We consider here a certain class of groupoids obtained via an equivalence relation (the so-called subgroupoids of pair groupoids). We generalize to Haar systems in these groupoids some results related to entropy and pressure which are well known in thermodynamic formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the a priori probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as well as of pressure for transverse functions. Moreover, we explore the relation between quasi-invariant probabilities and transverse measures. Some of the general results presented here are not for continuous modular functions but for the more general class of measurable modular functions.

In this paper we study topological cocycles for minimal homeomorphisms on a compact metric space. We introduce a notion of an essential range for topological cocycles with values in a locally compact group, and we show that this notion coincides with the well-known topological essential range if the group is abelian. We then define a regularity condition for cocycles and prove several results on the essential ranges and the orbit closures of the skew product of regular cocycles. Furthermore, we show that recurrent cocycles for a minimal rotation on a locally connected compact group are always regular, assuming that their ranges are in a nilpotent group, and then their essential ranges are almost connected.

This paper shows that any symplectic diffeomorphism $\mathbf{F} :(\Sigma^{2n},\eta) \to (\Sigma^{2n},\eta)$ can be embedded as a subsystem of a Liouville-integrable Hamiltonian flow on some symplectic manifold. If F is real-analytic, then the flow can be chosen to be real-analytic, but it is Liouville integrable with smooth first integrals. Examples are constructed of integrable, volume-preserving Hamiltonian flows on Poisson manifolds whose metric entropy with respect to the volume form is positive. Completely integrable Hamiltonian flows on a symplectic manifold are constructed which have positive metric entropy with respect to an invariant probability measure that is absolutely continuous with respect to the canonical volume form.

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

In this paper, we point out a close relationship between two standard classical problems in mechanics which have coexisted in textbooks for many decades: (1) the pendulum whose suspension point executes fast periodic motion along a given curve; and (2) the skate (known also as the Prytz planimeter, or the ‘bicycle’). More generally, we deal with dynamical systems subjected to rapidly oscillating forcing. Examples include: charged particles in rapidly oscillating electromagnetic fields, in particular the Paul trap; particles in an acoustic wave; a bead sliding on a rapidly vibrating hoop. It turns out that the averaged systems of such kind are approximated by a non-holonomic system. The holonomy turns out to have a transparent geometrical or physical interpretation. For the example of a particle in an acoustic wave the holonomy is directly proportional to the speed of the vibration-induced drift.

The theory of Hubbard trees provides an effective classification of nonlinear postcritically finite polynomial maps in ℂ. This note extends the classification to maps from a finite union of copies of ℂ to itself. Holomorphic proper maps on a finite union of copies of ℂ which are postcritically finite and nowhere linear can be characterized by a ‘forest’ made up of one tree for each copy of the set of complex numbers.

Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the second author to this restricted setting.

We study the local connectedness of the Julia set of the family $f_b(z)=z^m(z^n-b)$. Assuming that the Julia set is connected and depending on whether $m=1$ or $m>1$, a sufficient condition for the local connectivity of the Julia set is obtained. This complements the known results of Yoccoz–Hubbard–Branner for $m=n=1$.

We show that for a locally free $C^{\infty }$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real-valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys.27(2007), 1719–1735] concerning the Heisenberg groups.

Chaotic and ergodic properties are discussed for various subclasses of cylindric billiards. A common feature of the studied systems is that they satisfy a natural necessary condition for ergodicity and hyperbolicity, the so-called transitivity condition. The relation of our discussion to former results on hard ball systems is twofold. On the one hand, by slight adaptation of the proofs we may discuss hyperbolic and ergodic properties of 3 or 4 particles with (possibly restricted) hard ball interactions in any dimensions. On the other hand, a key tool in our investigations is a kind of connected path formula for cylindric billiards, which together with the conservation of momenta gives back, when applied to the special case of hard ball systems, the classical connected path formula.