We give an example of a strictly positive analytic reparametrization (or time change) of an irrational flow on \mathbb{T}^3 that is mixing. As an immediate application we obtain perturbations of completely integrable Hamiltonian systems that display many invariant tori on which the restricted dynamics is mixing.

Soit E\subset \mathbb{C} un compact de capacité logarithmique positive. Supposons que pour tout polynôme P\not=\text{id} on a P^{-1}(E)\not=E. Alors pour tout polynômes non constants f et g vérifiant f^{-1}(E)=g^{-1}(E), on a f=g, c'est-à-dire que E est un ensemble d'unicité pour les polynômes. La preuve de ce résultat utilise la théorie de systèmes dynamiques holomorphes en une et plusieurs variables.

The entropy for actions of a finitely-generated nilpotent group G is investigated. The Pinsker algebra of such actions is described explicitly. The systems with completely positive entropy are shown to have a sort of ‘asymptotic independence property’, just as in the case of \mathbb{Z}^d-actions. The invariant partitions are used to prove that the property of completely positive entropy is equivalent to the property of the K-system (the property of the existence of special ‘good’ partitions). A complete spectral characterization of K-systems is given. A construction of examples of completely positive non-Bernoullian actions of general countable nilpotent groups and a class of solvable groups is presented.

In this paper, we study the display of weak mixing by reparameterized linear flows on the torus \mathbb{T}^d, d\geq 2. We show that if the vector of the translation flow is Liouvillian (i.e. well approximated by rationals), then for a residual set of time change functions in the C^\infty topology, the reparameterized flow is weak mixing. If this is not the case, i.e. if the vector of the linear flow is Diophantine, it follows from a result of Kolmogorov on the two torus, and its generalization to any dimension by Herman, that any C^\infty reparameterization of the flow is C^\infty conjugate to a linear flow. More generally, in any given class of differentiability C^r for the time change function \phi, we give the optimal arithmetical condition on the vector of the translation flow that guarantees the existence of a residual set in the C^r topology of weak mixing reparameterizations. In the real analytic case, the optimal arithmetical condition for the generic display of weak mixing under time change is also given.

As a consequence of our results on reparameterizations of Liouvillian linear flows, we obtain that an aperiodic smooth flow on the two-dimensional torus is in general weak mixing. We also deduce the existence on the torus of analytic diffeomorphisms that are rank one and weak mixing.

We refine the transformation to smooth normal form for an analytic family of vector fields in the neighbourhood of a saddle-node. This refinement is very powerful and allows us to prove the finite cyclicity of families of graphics (‘ensembles’) occurring inside analytic families of vector fields. In [ZR1] and [ZR2] it is used to prove the finite cyclicity of graphics through a nilpotent singular point of elliptic type. Several examples are presented: lips, graphics with two subsequent lips, graphics with a nilpotent point of elliptic type and a saddle-node. We also discuss the bifurcation diagram of limit cycles for a graphic in the lips.

Hyperbolic sets are robust under perturbations: they persist on an open set of the parameter space. In this paper we investigate the boundary of this open set. Generalizing the theory of fixed points we define saddle-node bifurcations for hyperbolic sets K with one-dimensional unstable directions. In this bifurcation the geometrical splitting of the tangent space is preserved but the expansion in the unstable direction degenerates near a periodic orbit. The compact set K can be followed on a closed half-space bounded by a codimension-one manifold \mathcal{O}^0. On \mathcal{O}^0 the saddle-node bifurcation occurs. On one side of \mathcal{O}^0, K is hyperbolic and on the other side, it has disappeared.

We consider cocycles with negative Lyapunov exponents defined over a hyperbolic dynamical system. It is well known that such systems possess invariant graphs and that under spectral assumptions these graphs have some degree of Hölder regularity. When the invariant graph has a slightly higher Hölder exponent than the a priori lower bound on an open set (even on just a set of positive measure for certain systems), we show that the graph must be Lipschitz or (in the Anosov case) as smooth as the cocycle.

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.

We consider the collapse of a saddle of a horseshoe and a sink via a critical saddle-node bifurcation. In this way one obtains a saddle-node horseshoe. We prove that there is an open set of arcs of diffeomorphisms \{f_\mu\}_{\mu \in I} unfolding generically, say at \mu_0, a saddle-node horseshoe with Hausdorff dimension arbitrarily close to 1/2 so that there is an interval (\mu_0, \mu_0+\varepsilon) in the parameter line such that every diffeomorphism f_\mu, \mu\in (\mu_0, \mu_0+\delta), has a tangency.

Let T:J \to J be an expanding rational map of the Riemann sphere acting on its Julia set J and f:J\to \mathbb{R} denote a Hölder continuous function satisfying f(x) > \log|T^\prime(x)| for all x in J. Then for any point z_0 in J define the set D_{z_0}(f) of ‘well-approximable’ points to be the set of points in J which lie in the Euclidean ball

B\bigg(y,\exp\bigg(-\sum_{i=0}^{n-1} f(T^iy)\bigg)\bigg)

for infinitely many pairs (y,n) satisfying T^n(y)=z_0. In our 1997 paper, we calculated the Hausdorff dimension of D_{z_0} (f). In the present paper, we shall show that the Hausdorff measure \mathcal{H}^s of this set is either zero or infinite. This is in line with the general philosophy that all ‘naturally’ occurring sets of well-approximable points should have zero or infinite Hausdorff measure.

We investigate the iterative dynamics of symplectic piecewise-affine elliptic rotation maps associated with matrices with non-integer entries, \big(\begin{smallmatrix} 0 & -1 \\ 1 & a \end{smallmatrix}\big), where -2 < a < 2. We explain how their singularity and periodicity structures develop in general. In a special case, a = \pm \sqrt 2, we completely determine the orbit structure, the singularity structure, the invariant fractal and the ergodicity on the invariant fractal with respect to its Hausdorff measure. We also report an example of a period tripling cascade that arises from this system.

Let X be a compact, connected subset of a smooth manifold. Suppose that X admits a codimension d lamination, d\ge 2. Let x,y\in X and \epsilon>0. There exists a sequence x=x_0,\dotsc,x_k=y in X such that the sum of the dth powers of the distances between successive points is less than \epsilon. We discuss two proposed applications of this result.

We prove the existence of periodic orbits for steady real-analytic Euler flows on all Riemannian solid tori. By using the correspondence theorem from part I of this series, we reduce the problem to the Weinstein Conjecture for solid tori. We prove the Weinstein Conjecture on the solid torus via a combination of results due to Hofer et al and a careful analysis of tight contact structures on solid tori.

We consider intermittency in one-parameter families of unimodal maps, induced by saddle node and boundary crisis bifurcations. In these bifurcations either a periodic orbit or a periodic interval disappears to give rise to chaotic bursts. We prove asymptotic formulae for the frequency with which orbits visit the region previously occupied by the attractor. For this, we extend Pianigiani's results on conditionally invariant measures for the logistic family to more general families.

We discuss results on engaging actions of a lattice \Gamma in a higher rank simple group G on a compact manifold M. An action is engaging if there is no loss of ergodicity in passing to lifts of the action on finite covers of M.

Suppose \mathbb{R}-rank (G)\geq 2 and \Gamma<G is a lattice. Let \Lambda be the group of lifts of the \Gamma action on M to the universal cover of M. Assume the \Gamma action on M is measure preserving and engaging. We show that the image of \Lambda under any linear representation \sigma is \mathfrak{s}-arithmetic. Also, associated to each \sigma we have a measurable quotient of the \Gamma action on M; this measurable quotient is a generalized affine action on a double coset space. Furthermore, the pushforward of the invariant measure on M is a Lebesgue measure on the quotient. The fundamental group of this quotient is closely related to the image of \pi_1(M) under \sigma.

We prove the following result in commutative harmonic analysis.

THEOREM 1. Suppose\muis a probability measure on the dual\hat Gof a countable Abelian groupG, Eis a symmetric subset ofGand\sum_{g \in E-E, g \neq 0} |\widehat{\mu} (g)| < \delta < 1/3.Then there is anf \in C^+(\hat G)such that\|f-1\| < \tfrac{3}{2}\delta, (\widehat{f\mu})(g)=0for allg\in E\backslash \{0\}and(\widehat{f\mu})(0)=1.

This has the following corollary which leads to simple proofs and improvements of many of the results in a recent paper of Bergelson et al.

THEOREM 2.Ifg \mapsto U_gis a unitary action of a discrete Abelian groupGon a Hilbert space{\cal H}and\{g_i\} \subset Gis a sequence such that both\{g_i\}and\{2g_i\}are mixing for this action then for allx\in{\cal H}and\epsilon>0there are anx'\buildrel\epsilon\over\sim xand an IP-set\Gammagenerated by a subsequence of\{g_i\}such that the vectors\{U_gx':g\in\Gamma\}are pairwise orthogonal.

We also show that even in the case of a \mathbb{Z}-action (i.e. a single unitary operator U) the hypothesis \{g_i\} mixing alone is not sufficient for the conclusion of Theorem 2. Finally we give conditions on G and on the spectrum of the action under which that assumption is sufficient.

In this paper we consider some algebraic structure of groups of real-analytic diffeomorphisms of the circle.

We construct a diffeomorphism preserving a non-hyperbolic measure whose pointwise dimension does not exist almost everywhere. In the one-dimensional case we also show that such diffeomorphisms are typical in certain situations.

We investigate joinings of strongly irreducible totally non-symplectic Anosov {\mathbb Z}^k, k\ge 2 actions by toral automorphisms. We show that the existence of a non-trivial joining has strong implications for these actions, in particular, that the restrictions of the actions to a finite index subgroup of {\mathbb Z}^k are conjugate over {\mathbb Q}. We also obtain a description of the joining measures modulo the classification of zero entropy measures for the actions.

Moser's celebrated theorem guarantees that every diffeomorphism of a closed manifold can be isotoped to a volume-preserving one. We show that this statement cannot be extended into the contact category: some connected components of contactomorphism groups contain no volume-preserving maps. Thus, the dissipation of volume appears for purely topological reasons. This phenomenon can be considered from different viewpoints: geometric (isometric action of the contact mapping class group on the moduli space of contact forms), topological (action in symplectic homology) and dynamical (propagation of trajectories for symplectic maps). We define a numerical invariant—a contact Lyapunov exponent—which leads to a quantitive version of the above-mentioned result.