We prove the existence of a non-trivial homoclinic orbit on a Riemannian manifold (possibly non-compact), for Hamiltonian systems of the second order of the form:

where the potential V is T-periodic in the time variable.

For an arbitrary non-renormalizable unimodal map of the interval, f: I → I, with negative Schwarzian derivative, we construct a related map F defined on a countable union of intervals Δ. For each interval Δ, F restricted to Δ is a diffeomorphism which coincides with some iterate of f and whose range is a fixed subinterval of I. If F satisfies conditions of the Folklore Theorem, we call f expansion inducing. Let c be a critical point of f. For f satisfying f″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds for f: the ω-limit set of Lebesgue almost every point is the interval [f2, f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.

In this article, we prove a preparation theorem for functions which admit a certain type of expansion called Chebychev expansion. Taylor expansions are particular cases of Chebychev expansions. The result is based on an approach essentially different from those used for the classical preparation theorems. It has applications in bifurcation theory of vector fields.

In this paper we consider almost everywhere convergence and divergence properties of various ergodic averages. A general method is given which can be used to construct averages for which a.e. convergence fails, and to show divergence (and in some cases ‘strong sweeping out’) for large classes of ergodic averages. We also show that there are sequences with the gaps between successive terms converging to zero, but such that the Cesaro averages obtained by sampling a flow along these sequences of times converge a.e. for all f∈L1(X).

A distortion theory is developed for S-unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of S-unimodal maps is classified according to a distortion property, called the Markov-property.

We establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold a C∞ Riemannian metric whose geodesic flow is Bernoulli.

For α ∈ (0, 1), let Γα denote the middle-α Cantor set contained in the interval [0,1]. Let denote the set of all t∈[−1,1] such that Γα ∩(Γα+t)is a single point. Both a geometric and a symbolic description of each is presented. The symbolic description of each will be as a shift invariant subset of the one-sided two shift that is determined by inequalities. An equivalence relation is defined on the one-sided two shift and then a linear order relation is defined on the equivalence classes. This order relation on the equivalence classes describes how the sets shrink as a decreases from ⅓ to 0.

Let W⊥ be the class of all ergodic measure-preserving transformations (systems), which are disjoint from every weakly mixing system. Let M(W⊥) be the class of multipliers for W⊥; i.e. the class of all systems (X, μ, T) in W⊥ such that for every system (Y, ν, T) ∈ W⊥ and every ergodic joining λ of X and Y, the system (X × Y, λ, T × T) is also in W⊥. Well known results on disjointness show that the class D of ergodic distal systems, is a subclass of M(W⊥). Thus one has D ⊂ M(W⊥) ⊂ W⊥. Glasner and Weiss have shown that D ≠ W⊥. The purpose of this paper is to also show that D ≠ M(W⊥). The question whether M(W⊥) = W⊥ remains open.

By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling map h on the circle T. In particular, a connected and locally connected Julia set can be considered as a topological factor T/ ≈ of T with respect to a special h -invariant equivalence relation ≈ on T, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α → α from T onto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point α, they gave a topological classification of the abstract Julia sets T/ α. It turns out that T/ α contains simple closed curves iff the point α has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.

We describe the behaviour of the basin of attraction of the attracting periodic points which appear near a non-degenerate tangential homoclinic point of a dissipative saddle fixed point for one-parameter families of planar diffeomorphisms. This behaviour depends on certain relations between the eigenvalues of the saddle point and on the geometry of the tangency.

We generalize a result of Bourgain and a result of Huang. We also give a positive solution to A. Bellow's question: the a.e. convergence of the averages for σn = 1/n. On the other hand, we establish a sufficient and necessary condition for random sets in Z+ with asymptotic density zero which almost surely satisfy a mean ergodic theorem.

The topological entropy of a transformation is expressed in terms of partitions of unity. The transverse entropy of a flow tangential to a foliation is defined and expresed in a similar way. The geometric entropy of a foliation of a Riemannian manifold is compared with the transverse entropy of its geodesic flow.

We show that the strongest stable foliations associated with the generators of a Cartan action on a compact infra-nilmanifold are invaraint under topological conjugacy. This has the corollary that a Cartan action on a compact infra-nilmanifold with constant exponents is smoothly conjugate to an affine action.

We study the spectrum of a transfer operator that gives a counterexample to the conjecture of Mayer. The effect of perturbations is considered.

We prove that an area-preserving twist map having an invariant curve, can be approximated by a twist map exhibiting a Birkhoff-Hénon attractor. This is done by showing that the invariant curve can be perturbed into a saddle-node cycle with criticalities and by using a recent result reported by Diaz, Rocha and Viana.

Let h be the Hausdorff dimension of the Julia set of a rational map with no nonperiodic recurrent critical points. We give necessary and sufficient conditions for h-dimensional Hausdorff measure and h-dimensional packing measure of the Julia set to be positive and finite. We also show that either the Julia set is the whole Riemann sphere or h < 2 and that if a rational map (not necessarily with no nonperiodic recurrent critical points!) has a rationally indifferent periodic point, then h > 1/2.

This paper uses an elementary variational argument to establish the existence of solutions heteroclinic to a pair of periodic orbits for a class of Hamiltonian systems including Hamiltonians of multiple pendulum type.

The variation vector fields through extremals of the variational principles of natural Lagrangian functions satisfy the equation of Jacobi type. By making use of the Jacobi equation we obtain the estimates of measure-theoretic entropy for natural Lagrangian systems without conjugate points.

Let T be the universal covering tree of a finite graph, G. By analogy with an open problem concerning negatively curved covering manifolds, Kaimanovich asked when two of the three natural measure classes on ∂T can coincide, the three measures being harmonic measure, the Patterson measure, and visibility measure. We provide an almost complete answer and discuss related issues. The answer is quite surprising in some cases.