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).

Let M be a complete Riemannian locally symmetric space of non-positive curvature and of finite volume. We show that there are only finitely many compact maximal flats in M of volume bounded by a given number. As a corollary in the case $M=\mathrm{SL}_n(\mathbb{Z})\backslash\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n$, we give a different proof of a theorem of Remak that for any $n\in \mathbb{N}$, there are only finitely many totally real number fields of degree n whose regulator is less than a given number.

A diffeomorphism f of a compact manifold M is called almost Axiom-A if it is hyperbolic in a neighborhood of some compact f-invariant set $\Omega$, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure and such that for every point in this set the stable and unstable leaves are ‘long enough’, then f admits either a probability Sinai–Ruelle–Bowen (SRB)-measure or a $\sigma$-finite SRB-measure.

Let $\sigma$ and $\tau$ be two measure-preserving transformations of a non-atomic probability space, and Cob$(\sigma)$, Cob$(\tau)$ be the sets of their measurable coboundaries. We show that if the group G generated by $\sigma$ and $\tau$ is nilpotent and acts ergodically, then the inclusion Cob$(\sigma)\subseteq\text{Cob}(\tau)$ implies that $\sigma=\tau^n$ for some $n\in\mathbb Z$. This fact cannot be extended to solvable G. For G virtually solvable, a detailed description of the relationship between $\sigma$ and $\tau$ satisfying the inclusion Cob$(\sigma)\subseteq\text{Cob}(\tau)$ is given. In this case $\sigma$ is a generalized power of $\tau$ and is isomorphic to some $\tau^n,\ n\in\mathbb Z$.

The proofs require some study of non-free measure-preserving actions of elementary amenable groups and their stabilizers. In particular, a version of the Rokhlin lemma for non-free measure-preserving actions admitting maximal stabilizers is given.

A singular-hyperbolic attractor for vector fields is a partially hyperbolic attractor with singularities (that are hyperbolic) and volume expanding central direction. The geometric Lorenz attractor is the most representative example of a singular-hyperbolic attractor. In this paper, we prove that if $\Lambda$ is a singular-hyperbolic attractor of a three-dimensional vector field X, then there is a neighborhood U of $\Lambda$ in M such that every attractor in U of a Cr vector field Cr close to X is singular, i.e. it contains a singularity. With this result we prove the following corollaries. There are neighborhoods U of $\Lambda$ (in M) and $\mathcal U$ of X (in the space of Cr vector fields) such that if n denotes the number of singularities of X in $\Lambda$, then $\#\{A\subset U:A$ is an attractor of $Y\in\mathcal U\}\leq n$. Every three-dimensional vector field Cr close to one exhibiting a singular-hyperbolic attractor has a singularity non-isolated in the non-wandering set. A singularity of a three-dimensional Cr vector field Y is stably non-isolated in the non-wandering set if it is the unique singularity of a singular-hyperbolic attractor of Y. These results generalize well-known properties of the geometric Lorenz attractor.

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).

Krieger's embedding theorem for $\mathbb{Z}$ mixing shifts for finite type (SFTs) is extended to the $\mathbb{Z}^2$ case. We prove that if X is a non-periodic subshift ($\sigma^vx=x\Rightarrow v=0\in\mathbb{Z}^2$) and Y is a $\mathbb{Z}^2$ square-filling mixing SFT, then there exists a homomorphism $X\to Y$. The proof is a construction which begins by constructing Voronoi tilings using techniques from Part I (S. Lightwood. Morphisms from non-periodic $\mathbb{Z}^2$ subshifts I: constructing embeddings from homomorphisms. Ergod. Th. & Dynam. Sys.23 (2003), 587–609.). A tiling by Delaunay polygons is derived from Voronoi tiling. The union of the boundaries of the Delaunay polygons (referred to as the Delaunay graph) is itself tiled by trees. Words painted on the thickened trees combine to form words on the thickened infinite Delaunay graph. It is the point and sole purpose of square-filling that words on such thickened infinite graphs will correspond to points in the target space.

Combined with the results from Part I, this gives us the $\mathbb{Z}^2$ extension of Krieger's Embedding Theorem: if X is a non-periodic subshift and Y is a $\mathbb{Z}^2$ square-filling mixing SFT, then there exists an embedding $X\hookrightarrow Y$ if and only if $h(X)<h(Y)$, where h denotes the $\mathbb{Z}^2$ entropy. The techniques developed here play a central role in the proof of an embedding theorem for general $\mathbb{Z}^2$ subshifts into square-filling mixing SFTs which will be carried out in a subsequent paper.

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.

If M is a compact or convex-cocompact negatively curved manifold, we associate to any Gibbs measure on T1M a quasi-invariant transverse measure for the horospherical foliation, and prove geometrically that this measure is uniquely determined by its Radon–Nikodym cocycle. (This extends the Bowen–Marcus unique ergodicity result for this foliation.) We shall deduce from this result some equidistribution properties for the leaves of the foliation with respect to these Gibbs measures. We use it also in the study of invariant measures for horospherical foliations on regular covers of M.

In the 1970s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system with a perturbation of size $\varepsilon$, the actions linked to the unperturbed Hamiltonian vary only by the order of $\varepsilon^b$ over a time of the order of $\exp (C\varepsilon^{-a})$ for some positive constants a, b and C, provided that the unperturbed Hamiltonian meets some generic transversality condition known as steepness. Among steep systems, convex or quasiconvex systems are easier to describe since the use of energy conservation allows the proof of exponential estimates of stability to be shortened. In this case, Lochak–Neishtadt and Poschel have independently obtained the stability exponents a = b = 1/2n for systems of n degrees of freedom—especially the time exponent (a) is expected to be optimal (see P. Lochak, J.-P. Marco and D. Sauzin. Preprint. Institut de Máthematique de Jussieu, 1999; J.-P. Marco and D. Sauzin. Preprint. Publ. Math. Inst. Hautes Etudes Science, 2001). Moreover, Lochak's study relies on simultaneous Diophantine approximation which gives a very transparent proof.

However, the proof in the steep case has rarely been studied since Nekhorochev's original work despite various physical examples where the model Hamiltonian is only steep. Here, we combine the original scheme with a simultaneous Diophantine approximation as in Lochak's proof. This yields significant simplifications with respect to Nekhorochev's reasoning: it also allows the exponents $a=b=(2n p_1\dotsb p_{n-1})^{-1}$ where $(p_1\dotsb p_{n-1})$ are the steepness indices of the considered Hamiltonian to be obtained. In the quasiconvex case, the steepness indices are all equal to one and we find the same exponents 1/2n as Lochak–Neishtadt and Poschel, whose results are thus generalized to the steep case.

We prove that for all rational functions f on the Riemann sphere and potential $-t\ln|f'|, t\ge 0$ all the notions of pressure introduced in Przytycki (Proc. Amer. Math. Soc.351(5) (1999), 2081–2099) coincide. In particular, we get a new simple proof of the equality between the hyperbolic Hausdorff dimension and the minimal exponent of conformal measure on a Julia set. We prove that these pressures are equal to the pressure defined with the use of periodic orbits under an assumption that there are not many periodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet–Eckmann condition. In Appendix A, we discuss the case t < 0.

For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Young's axioms (Lai-Sang Young, Ann. Math.147 (1998), 585–650), which imply exponential decay of correlations and the central limit theorem (CLT), a local CLT is proven. In fact, a unified version of the local CLT is found, covering, among others, the absolutely continuous and arithmetic cases. For planar Lorentz process with a finite horizon, this result implies (a) a local CLT and (b) recurrence. For the latter case (d = 2, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt (C. R. Acad. Sci. Paris Ser. 1 Math. 372(9) (1998), 837–842) and J.-P. Conze (Ergod. Th. & Dynam. Sys.19(5) (1999), 1233–1245) could already establish recurrence.

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.

It is well known that the action of a hyperbolic element (‘cat map’) of the modular group on the 2-torus has strong chaotic dynamical properties such as mixing and exponential decay of correlations. In this note we study stability of this behaviour with respect to kicks. Our approach is based on geometric group theory, and in particular on a new result on quasi-morphisms of the modular group.

In this paper, we prove a relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension two, robustly transitive diffeomorphisms are robustly ergodic. In higher dimensions, we work with partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. We define almost robust ergodicity and prove that in the three-dimensional case robustly transitive diffeomorphisms are approximated by almost robustly ergodic ones. In higher dimensions under the condition of the central Lyapunov exponents being of the same sign, we prove that robustly transitive diffeomorphisms are robustly ergodic.

For every point $\chi$ in the spectrum of the operator $(h(\delta)\xi)=\xi_{n+1}+\xi_{n-1}+\beta(\delta e^{2\pi\alpha ni}+\delta^{-1}e^{-2\pi\alpha ni})\xi_n$ on $\ell^2(\mathbb{Z})$ there exists a complex number x of modulus one such that the equation $\xi_{n+1}+\xi_{n-1}+\beta(x\delta e^{2\pi\alpha ni}+\bar{x}\delta^{-1}e^{-2\pi\alpha ni})\xi_n=\chi\xi_n$ has a non-trivial solution satisfying the condition $\limsup_{|n|\to\infty}|\xi_n|^{1/|n|}\le\delta^{-1}\beta^{-1}$ provided that $\beta,\delta>1$ and $\alpha$ satisfies the diophantine condition $\lim_{n\to\infty}|{\rm sin}\,\pi\alpha n|^{-{1/n}}=1$. The parameters $x\delta$ and $\chi$ are in the range of analytic functions which are defined on a Riemann surface covering the resolvent set of the operator h(1).

We deal with all the mappings $f_\lambda(z)=\lambda e^z$ that have an attracting periodic orbit. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ that do not escape to infinity under positive iterates of $f_\lambda$. Our ultimate result is that the function $\lambda\mapsto{\rm HD}(J_r(f_\lambda))$ is real-analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form $-t\log|F_\lambda'|$, where $F_\lambda$ is the natural map associated with $f_\lambda$ closely related to the corresponding map introduced in Urbański and Zdunik (2001). The formalism includes appropriately defined topological pressure, Perron–Frobenius operators, and geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron–Frobenius operators are quasicompact and that they embed into a family of operators depending holomorphically on an appropriate parameter, and we obtain several other properties of these operators. We prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for the topological pressure is independent of the set Jr(f), Bowen's formula also indicates that Jr(f) is the right set to deal with. We also study in detail the properties of quasiconformal conjugacies between the maps $f_\lambda$. As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and exponential decay of correlations.

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 investigate normalizers of full groups of a topological dynamical system by means of cocycles. Some of the investigations are conducted in connection with index maps, the centralizer of a topological dynamical system and groups of automorphisms of a C*-crossed product.

Let f and g be two commuting holomorphic self-maps of the open unit disc $\mathbb{D}$ in the complex plane with a common Wolff point $\tau\in\partial\mathbb{D}$: if these two maps agree at $\tau$ up to the third order, then $f\equiv g$.