First we describe the work of the first author leading to the conclusion that any property generic (in the weak topology) for measure-preserving bijections of a Lebesgue probability space is also generic (in the compact-open topology) for homeomorphisms of a compact manifold preserving a fixed measure. Then we describe the work of both authors in extending this result to non-compact manifolds, with modifications based on the ends of the manifold. These results can be thought of as generalizations of the original work which established genericity for the specific property of ergodicity (Oxtoby and Ulam, 1941) and subsequent work for other properties such as weak mixing (Katok and Stepin, 1970). The techniques used to obtain the titled theorem are also applied to related areas, such as fixed point theorems and chaos theory, and some new results are obtained.

In this paper we prove the existence of an entire transcendental function of arbitrarily slow growth which exhibits simple dynamics on its Fatou set.

We investigate bifurcations in the chain recurrent set for a particular class of one-parameter families of diffeomorphisms in the plane. We give necessary and sufficient conditions for a discontinuous change in the chain recurrent set to occur at a point of heteroclinic tangency. These are also necessary and sufficient conditions for an \Omega-explosion to occur at that point.

We study a class of open chaotic dynamical systems. Consider an expanding map of an interval from which a few small open subintervals are removed (thus creating ‘holes’). Almost every point of the original interval then eventually escapes through the holes, so there can be no absolutely continuous invariant measures. We construct a so-called conditionally invariant measure that is equivalent to the Lebesgue measure. Our measure is unique and naturally generates an invariant measure, which is singular. These results generalize early work by Pianigiani, Yorke, Collet, Martinez, and Schmidt, who studied similar maps under an additional Markov assumption. We do not assume any Markov property here and use ‘bounded variation’ techniques rather than Markov coding. Our results supplement those of Keller, who studied analytic interval maps with holes by using different techniques.

We consider open sets of transformations in a manifold M, exhibiting non-uniformly expanding behavior in some forward invariant domain U\subset M. Assuming that each transformation has a unique Sinai–Ruelle–Bowen (SRB) measure in U, and some general uniformity conditions, we prove that the SRB measure varies continuously with the dynamics in the L1-norm. We also describe a concrete application.

Except for the painful beginning and end, well-deserved good fortune accompanied Moser all his life. Accomplishment and recognition came to him early and lastingly. He was the first recipient of the G. D. Birkhoff Prize of the AMS and SIAM. He was honored by a Wolf Prize, medals were showered on him, academies vied to elect him as their member. He was an outstanding expositor, much in demand. He delivered the Gibbs lecture of the AMS, the Pauli lectures of the ETH, the Hardy lectures of Cambridge University, the Hermann Weyl lectures of the Accademia Lincei, and the von Neumann lecture of SIAM. He was president of the IMU for three years.

He was fortunate in having many good students, with whom he continued close scientific and personal relations. He loved his family fiercely, two daughters and a stepson, six grandchildren and charming and talented sons-in-law.

According to an old Roman proverb, ‘what doesn't destroy me makes me stronger’. This fits Jürgen, for in his early youth he was exposed to deadly danger, but survived the stronger for it. In 1938, the ten year old Jürgen was offered a chance to enroll in an elite boarding school for future leaders, but his parents who saw the Nazi's plan were able to extricate him. Jürgen's father was a neurologist, and as such was sometimes called upon to declare someone as ‘unfit to live’ according to the Nazi eugenics program. Jürgen's father, courageously, always refused to issue such certifications.

Let f be a rational function with a fixed point z_0 of multiplicity m+1. Then there are m invariant components of the Fatou set, called immediate parabolic basins, where the iterates of f tend to z_0, and each immediate parabolic basin contains at least one critical point of f. A function \phi holomorphic and univalent near z_0 with \phi(z_0)=0 such that h:=\phi\circ f\circ\phi^{-1} satisfies h(z)=z-z^{m+1}+\alpha z^{2m+1}+O(z^{2m+2}) as z\to 0 for some \alpha\in\mathbb{C} also exists. This number \alpha is invariant under holomorphic changes of coordinates. We show that if each immediate parabolic basin at z_0 contains exactly one critical point of f, and if this critical point is simple, then \textrm{Re}\alpha\leq \frac14m+\frac12. We also discuss the case that the critical points in the immediate parabolic basins are multiple and establish an upper bound for the real part of \alpha depending on m and the multiplicities of the critical points contained in the parabolic basins.

Jürgen Moser (1928–1999) was one of the most accomplished mathematicians of the second half of the 20th century and his work had a major impact in broad areas of analysis, especially partial differential equations and dynamical systems, and geometry. In this article we discuss only his contributions to dynamics and closely related areas. We feel that the best way to make the reader understand and appreciate the impact of Moser's work is to discuss it within the framework of some major trends in the development of dynamical systems during the last century. We apologize in advance for omitting many important references, including some key original work. These can be found in secondary sources to which we refer for detailed accounts of various areas. Needless to say, we included in our references all papers by Moser relevant to our discussion. For brief surveys of Moser's work in all areas see [45,91].

The leading theme of virtually all of Moser's work in dynamics is the search for elements of stable behavior in dynamical systems with respect to either initial conditions or perturbations of the system.

We construct an algebraic method for detecting bifurcations of discrete dynamical systems using the Conley index theory. We define the transition matrix pair for discrete semidynamical systems, which detects bifurcations of connecting orbits. As an application, we give a sufficient condition for the occurrence of homoclinic and heteroclinic tangencies by showing that the occurrence of a heterodimensional cycle in the projectivization of a dynamical system implies the tangency of the original system.

We provide new non-trivial quantitative information on the behavior of Poincaré recurrence. In particular we establish the almost everywhere coincidence of the recurrence rate and of the pointwise dimension for a large class of repellers, including repellers without finite Markov partitions.

Using this information, we are able to show that for locally maximal hyperbolic sets the recurrence rate possesses a certain local product structure, which closely imitates the product structure provided by the families of local stable and unstable manifolds, as well as the almost product structure of hyperbolic measures.

Soient f un endomorphisme holomorphe de degrée plus grand que 2, et \mu sa mesure d'équilibre. Alors (f,\mu) posséde la propriété de Bernoulli.

We prove that if (M,\omega) is a connected and compact four-dimensional symplectic manifold, there exist three open sets U_1, U_2, U_3 of {\rm Diff}^1_{\omega}(M) (for the C^1 topology) such that:

1. U_1\cup U_2\cup U_3 is dense in {\rm Diff}^1_{\omega}(M);

2. f\in U_1 if and only if f is Anosov and transitive;

3. f\in U_2 if and only if f is partially hyperbolic; and

4. f\in U_3 if and only if f has a stable completely elliptic periodic point.

For any stationary process X=X_1,X_2,\dotsc with shift map \sigma, and T an invertible, measure-preserving flow on a probability space, consider the random walk on a random scenery \hat{T}_X(x,\omega)=(\sigma(x),T_{x_0}(\omega)). We prove that if X satisfies a certain property and T has positive entropy, then the number of names in the Vershik metric v_n increases superpolynomially in n. This fact allows us to prove that a class of smooth maps are not standard in the sense of Vershik.

For any \varepsilon > 0, we construct an explicit smooth Riemannian metric on the sphere S^n, n \geq 3, that is within \varepsilon of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is \varepsilon-dense in the unit tangent bundle. Moreover, for any \varepsilon > 0, we construct a smooth Riemannian metric on S^n, n \geq 3, that is within \varepsilon of the round metric and has a geodesic for which the complement of the closure of the corresponding orbit of the geodesic flow has Liouville measure less than \varepsilon.

We show that in a topological dynamical system (X,T) of positive entropy there exist proper (positively) asymptotic pairs, that is, pairs (x,y) such that x\not= y and \lim_{n\to +\infty} d(T^n x,T^n y)=0. More precisely we consider a T-ergodic measure \mu of positive entropy and prove that the set of points that belong to a proper asymptotic pair is of measure one. When T is invertible, the stable classes (i.e. the equivalence classes for the asymptotic equivalence) are not stable under T^{-1}: for \mu-almost every x there are uncountably many y that are asymptotic to x and such that (x,y) is a Li–Yorke pair with respect to T^{-1}. We also show that asymptotic pairs are dense in the set of topological entropy pairs.

The finite Pfaff lattice is given by a commuting Lax pair involving a finite matrix L (zero above the first subdiagonal) and a projection onto sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time parameters t_1,t_2,\dotsc, after conjugation by a diagonal matrix. The sequence of polynomial \tau-functions, solving the problem, belongs to an intriguing chain of subspaces of Schur polynomials, associated to Young diagrams, dual with respect to a finite chain of rectangles. Also, this sequence of \tau-functions is given inductively by the action of a fixed vertex operator.

As an example, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial.

For a closed lamination on the unit circle invariant under z\mapsto z^d we prove an inequality relating the number of points in the ‘gaps’ with infinite pairwise disjoint orbits to the degree; in particular, this gives estimates on the cardinality of any such ‘gap’ as well as on the number of distinct grand orbits of such ‘gaps’. As a tool, we introduce and study a dynamically defined growing tree in the quotient space. We also use our techniques to obtain for laminations an analog of Sullivan's no wandering domain theorem. Then we apply these results to Julia sets of polynomials.

Connections are established between mixing or ergodic properties of maps on the one hand, and the convergence of the iterates of the map, or of the empirical measures of the iterates, to a constant measure-valued map, on the other. The uniqueness of an absolutely continuous ergodic measure can also be verified via the convergence. The technique helps to identify ergodic and mixing pairs and verify the uniqueness in specific examples.

Let \{a_k\}_{k\geq0} be a sequence of complex numbers. We obtain the necessary and sufficient conditions for the convergence of n^{-1}\sum_{k=0}^na_kT^kx for every contraction T on a Hilbert space H and every x \in H. It is shown that a natural strengthening of the conditions does not yield convergence for all weakly almost periodic operators in Banach spaces, and the relations between the conditions are exhibited. For a strictly increasing sequence of positive integers \{k_j\}, we study the problem of when n^{-1}\sum_{j=1}^nT^{k_j}x converges to a T-fixed point for every weakly almost periodic T or for every contraction in a Hilbert space and not for every weakly almost periodic operator.

Minimization arguments are applied to a renormalized length functional to establish the existence of heteroclinic, homoclinic and chaotic geodesics on a two-dimensional torus satisfying some geometrical conditions.