In this article we give a necessary and sufficient condition for the conjugacy of the expanding maps on the circle S1 to be differentiable.

We study invariant measures for a continuous function which maps a real interval into itself. We show that the ratio of the measures of the two subintervals into which it is divided by a fixed point is constrained by the the set of periods of periodic points. As a consequence of this we give a new forcing relation between periodic points.

We study several notions of positivity for a class of real-valued functions of several variables that includes the Laurent polynomials. We show that Handelman's positivity condition is characterized by boundedness of the associated Legendre transformation, or boundedness of the entropy function, while another notion of positivity here introduced characterizes the mapping property of the Legendre transformation derived by Marcus and Tuncel for beta functions. Several examples are given to distinguish the various notions of positivity.

We present a new proof of a strengthened version of Anosov's multiphase averaging theorem, originally stated for systems of ODEs with slow variables evolving in Rm and fast variables evolving on a smooth immersed manifold. Our result allows the fast variables to belong to an arbitrary smooth compact Riemannian manifold, and the vector field to have only Sobolev regularity. This is accomplished using normal form techniques adapted to a slightly generalized version of the DiPema-Lions theory of generalized flows for ODEs.

For a fixed irrational α > 0 we say that probability measure-preserving transformations S and T are α-equivalent if they can be realized as cross-sections in a common flow such that the return time functions on the cross-sections both take values in {1, 1 +α} and have equal integrals. Similarly we call two flows F and G α-equivalent if F has a cross-section S and G has a cross-section T isomorphic to S and again both the return time functions take values in {1, 1 + α} and have equal integrals. The integer kα(S), equal to the least positive such such that exp2πikα-1 belongs to the point spectrum of S, is an invariant of α-equivalence.

We obtain a characterization of a-equivalence as a particular type of restricted orbit equivalence and use this to prove that within the class of loosely Bernoulli maps ka(S) together with the entropy h(S) are complete invariants of α-equivalence. There is a corresponding a-equivalence theorem for flows which has as a consequence, for example, that up to an obvious entropy restriction, any weakly mixing cross-section of a loosely Bernoulli flow can also be realized as a cross-section with a {1,1 + α}-valued return time function.

For the proof of the α-equivalence theorem we develop a relative Kakutani equivalence theorem for compact group extensions which is of interest in its own right. Finally, an example of Fieldsteel and Rudolph is used to show that in general kα(S) is not a complete invariant of α-equivalence within a given even Kakutani equivalence class.

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.

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.

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.

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.

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 G be a locally compact σ -compact group with right Haar measure λ. A sequence {μn} of probabilities on G is called ergodic if for every f ∈ L1(G, λ) and t ∈ G we have ‖μn* (f − δt* f) ‖1 → 0. If T (t) is a bounded continuous representation of G by linear operators in a Banach space X, we define the μ,-average of T(t) by .

Let T be a measure-preserving transformation of the Lebesgue probability space (X, F, μ). We call (X, Fμ, T) a dynamical system. Bourgain has proven the following theorem.

Due to an editorial omission the full acknowledgments to Maria Carvalho's paper Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms (Volume 13: issue 1, p. 44) was not printed. These missing lines are printed below.

Acknowledgment. I was most fortunate to have Professor Ricardo Mané as my PhD supervisor. I am grateful to him for bringing me to this subject, his invaluable guidance and encouragement. I am also thankful to L. Diaz, J. Rocha and M. Viana for helpful discussions.