The untimely death of Michael Robert Herman in November 2000 deprived the scientific community of one of its deepest mathematical minds, who had a profound impact on the theory of dynamical systems over the last 30 years.

Born in New York, he was educated in France. He was a student at École Polytechnique before being one of the first members of the Centre de Mathématiques created there by Laurent Schwartz. For more than 20 years, his seminar had a major influence worldwide and was the main vector of the development of the theory of dynamical systems in France. All of his students remember with thankfulness and emotion the passion with which he led them into the wonderful mathematical world. He maintained through the years strong connections with the Instituto de Matemática Pura e Aplicada in Rio de Janeiro.

His interests covered most aspects of the modern theory of dynamical systems and much beyond that, from economics to arts and philosophy. However, it is fair to say that from the start the so-called small divisors problems, related in particular to the stability of quasiperiodic motions, were closest to his heart. His epoch-making theorem on the linearization of circle diffeomorphisms [1–4], his two volumes [5, 6] on invariant curves for twist diffeomorphisms, which are still the standard reference 20 years later, his very many deep contributions on the existence and geometry of invariant tori all bear witness to that interest [7–10].

We show that if $\varphi$ is a lacunary Fourier series and the equation $\psi (x) -\psi (x + \alpha) = \varphi(x), x \bmod 1$ has a measurable solution $\varphi$, then in fact the equation has a solution in L2.

For any ergodic endomorphism of a non-atomic probability space and any Borel generating set of a locally compact Polish (LCP) amenable group, we construct a measurable function taking values in the generating set so that the skew product generated is ergodic.

We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is expressed by a geometric condition on the cocycle's behaviour over periodic points and associated homoclinic orbits. It allows us to prove that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Hölder continuous potentials), all Oseledets subspaces are one-dimensional. Moreover, the complement of this subset has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters.

This improves previous results of Bonatti, Gomez–Mont and Viana where it was shown that some Lyapunov exponent is non-zero, in a similar setting and also for an open dense subset.

This is a partial survey of various developments of the classical KAM theory in dynamical systems to infinite-dimensional lattice models and Hamiltonian PDEs. Particular emphasis is given on Nekhoroshev stability and growth of higher Sobolev norms. Some new results are also presented.

We establish rather general and simple theorems implying, among other things, the pseudo-(un)stable manifold theorem, Sternberg's theorem on smooth conjugacy between hyperbolic germs of maps or vector fields, and results of Fenichel, Hirsch, Pugh and Shub on existence, uniqueness and structural stability of stable or unstable manifolds at compact invariant manifolds. The Perron–Irwin approach via sequence spaces plays a crucial role.

We prove that a Riemannian metric on the 2-sphere or the projective plane can be C2 approximated by a $C^\infty$ metric whose geodesic flow has an elliptic closed geodesic.

We consider systems that have some hyperbolicity behavior and which preserve conformal structures on the stable and unstable bundles. We show that two such systems that are topologically conjugate are smoothly conjugate. This is somewhat more general than a conjecture of the author in 2002. Related results have also been obtained by Kalinin and Sadovskaia.

In this paper we consider a quasi-linear approximation for the 3D Euler system as well as for the Navier–Stokes system in our previous paper Moscow Math. J.1(3) (2001), 381–388). We define finite-dimensional versions of the approximation and study properties of their solutions.

In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space X admits a Lyapunov 1-form $\omega$ lying in a prescribed Čech cohomology class $\xi\in\check H^1(X;\mathbb{R})$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem by Conley about the existence of Lyapunov functions.

We present an overview and some new applications of the approximation by conjugation method introduced by Anosov and Katok more than 30 years ago (Trans. Moscow Math. Soc.23 (1970), 1–35). Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi (Asterisque49 (1977), 37–59) and continuing with exotic invariant sets of rational maps of the Riemann sphere (J. London Math. Soc. (2) 34 (1986), 375–384) and the construction of invariant tori with non-standard and unexpected behavior in the context of KAM theory (Pitman Research Notes Mathematical Series 243 (1992); Proc. Int. Congr. Mathematicians (Berlin, 1998) Vol. 11, 797–808). Recently the method has been experiencing a revival. Some of the new results presented in the paper illustrate variety of uses for tools available for a long time, others exploit new methods, in particular the possibility of mixing in the context of Liouvillean dynamics discovered by the first author (Ergod. Th. & Dynam. Sys.22 (2002) 437–468; Proc. Amer. Math. Soc.130 (2002), 103–109).

V. I. Arnold (Petits dénominateurs et problèmes de stabilité du mouvement en mécanique classique et en mécanique céleste. Usp. Mat. Nauk.18 (1963), 91–192 (en russe)) a affirmé et partiellement démontré que, pour le modèle newtonien du Système solaire à $n\geq 2$ planètes dans l'espace, si la masse des planètes est suffisamment petite par rapport à celle du Soleil, il existe, dans l'espace des phases au voisinage des mouvements képlériens circulaires coplanaires, un sous-ensemble de mesure de Lebesgue strictement positive de conditions initiales conduisant à des mouvements quasipériodiques à 3n - 1 fréquences. Cet article détaille la démonstration que M. R. Herman a exposée de ce théorème (Démonstration d'un théorème de V. I. Arnold, Séminaire de Systèmes Dynamiques et manuscrits, 1998).

Un point matériel soumis à une force centrale attractive dérivant d'un potentiel possède une famille à un paramètre d'orbites périodiques circulaires. Le théorème de Bertrand affirme que si toutes les orbites voisines de ces orbites circulaires sont périodiques, le potentiel est newtonien (proportionel à 1/r, où r est la distance du point au centre attractif) ou élastique (en r2) (J. Bertrand. Comptes Rendus77 (1873), 849–853). Nous calculons, sur une idée initiale de Michael Herman et pour un potentiel central générique, les deux premiers invariants de Birkhoff du système le long des orbites circulaires ; puis nous montrons comment le théorème de Bertrand s'en déduit.

For any compact oriented surface $\Sigma$ we consider the group of diffeomorphisms of $\Sigma$ which preserve a given area form. In this paper we show that the vector space of homogeneous quasi-morphisms on this group has infinite dimension. This result is proved by constructing explicitly and for each surface an infinite family of independent homogeneous quasi-morphisms. These constructions use simple arguments related to linking properties of the orbits of the diffeomorphisms.

We construct examples of Gevrey non-analytic perturbations of an integrable Hamiltonian system which give rise to an open set of unstable orbits and to a special kind of symbolic dynamics. We find an open ball in the phase space, which is transported by the Hamiltonian flow from $-\infty$ to $+\infty$ along one coordinate axis, at a speed that we estimate with respect to the size of the perturbation. Taking advantage of the hyperbolic features of this unstable system, particularly the splitting of invariant manifolds, we can also embed a random walk along this axis into the near-integrable dynamics.

We construct a quotient Aubry set isometric to the unit interval. We also construct several related examples of Aubry sets.

We survey some recently obtained generic consequences of the existence of homoclinic tangencies in diffeomorphisms of surfaces. Among other things, it has been shown that they give rise to invariant topologically transitive sets with maximal Hausdorff dimension, that they prohibit the existence of various kinds of symbolic extensions and that they form an impediment to the existence of Sinai–Ruelle–Bowen (SRB) measures. The main new result described here, together with a positive answer to an as yet unproved conjecture of Palis, would prove that generically on surfaces, SRB measures only exist on uniformly hyperbolic attractors.

The so-called Herman–Światek theorem is generalized to holomorphic self-homeomorphisms of quasi-circles. This result implies an unpublished theorem of Michel Herman: If a Siegel disk or Arnold–Herman ring for a rational map has a boundary component, which is a quasi-circle containing a critical point, then the associated rotation number is Diophantine of exponent 2.

We consider Gevrey perturbations H of a completely integrable non-degenerate Gevrey Hamiltonian H0. Given a Cantor set $\Omega_\kappa$ defined by a Diophantine condition, we find a family of Kolmogorov–Arnold–Moser (KAM) invariant tori of H with frequencies $\omega\in \Omega_\kappa$ which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda$ of the invariant tori. This leads to effective stability of the quasi-periodic motion near $\Lambda$.

We study an analytic Hamiltonian system with two degrees of freedom, having the origin as an elliptic singularity. We assume that the full Birkhoff normal form exists and is divisible by its quadratic part, being indefinite. We show that under the Bruno condition and under the restriction to the zero energy surface, a real analytic transformation into a normal form exists. Such a normal form coincides with the restriction of the Birkhoff normal form to the zero energy surface up to an order as large as we want.