We consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28–30 October, 1981). IEEE, New York, 1981, pp. 364–375] from deterministic weight distributions (Dirac measures μ) to general ones. Given a probability measure μ which corresponds to the weight distribution of a link of a random graph, we form a positive linear operator Φμ (convolution) on distribution functions and then analyze a family of its exponents, with parameter λ, which corresponds to the connectivity of a sparse random graph. The operator 𝕋 relates the distribution F on the subtrees to the distribution 𝕋F on the node of the tree by 𝕋F=exp (−λΦμF). We prove that for every probability measure μ and every λ<e, there exists a unique globally attracting fixed point of the operator; the probability measure corresponding to this fixed point can then be used to compute the expected maximum-weight matching on a sparse random graph. This result is called the e-cutoff phenomenon. For deterministic distributions and λ>e, there is no fixed point attractor. We further establish that the uniqueness of the invariant measure of the underlying operator is not a monotone property of the average connectivity; this parallels similar non-monotonicity results in the statistical physics context.

We consider stationary ergodic processes indexed by or whose finite-dimensional marginals have laws which are absolutely continuous with respect to Lebesgue measure. We define an entropy theory for these continuous processes, prove an analogue of the Shannon–MacMillan–Breiman theorem and study more precisely the particular example of Gaussian processes.

We characterize convex cocompact subgroups of the mapping class group of a surface in terms of uniform convergence actions on the zero locus of the limit set. We also construct subgroups that act as uniform convergence groups on their limit sets, but are not convex cocompact.

This paper is about flow–orbit equivalence, a topological analog of even Kakutani equivalence. In addition to establishing many basic facts about this relation, we characterize the conjugacies of induced systems that can be extended to a flow–orbit equivalence. We also describe the relationship between flow–orbit equivalence and a distortion function of an orbit equivalence. We show that, if the distortion of an orbit equivalence is zero, then it is in fact a flow–orbit equivalence, and that the converse is true up to a conjugation by an element of the full group.

For a generic C1 expanding map of the circle, the Lyapunov maximizing measure is unique and fully supported, and has zero entropy.

A class of piecewise affine hyperbolic maps on a bounded subset of the plane is considered. It is shown that if a map from this class is sufficiently area-expanding then almost surely this map has an absolutely continuous invariant measure.

Let Π be an open period annulus of a plane analytic vector field X0. We prove that the maximal number of limit cycles which bifurcate from Π under a given multi-parameter analytic deformation Xλ of X0 is the same as in an appropriate one-parameter analytic deformation Xλ(ε), provided that this cyclicity is finite. Along the same lines, we also give a bound for the cyclicity of homoclinic saddle loops.

We construct an adic model for the Denjoy system (X,T) of finite double orbit number. It is based on the dual Ostrowski numeration system. By using this construction, we can determine the K-theory of associated with (X,T) in an effective way.

We develop a Ruelle–Perron–Fröbenius transfer operator approach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For Hölder continuous potentials not too far from constant, we prove that the transfer operator has a positive eigenfunction, which is piecewise Hölder continuous, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a non-lacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here.

Twist maps (θ1,r1)=f(θ,r) are considered in this paper, with no assumption on the periodicity of the map in θ. Under appropriate assumptions, the existence of infinitely many bounded (in r) complete orbits is proven. In particular, our results apply to the class of maps where λ>0 and no arithmetic condition has to be imposed on ω1/ω2.

We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being ‘small’ in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case—when Y is a finite set—this result is highly non-trivial. The result itself—called the absorption theorem—is a powerful and crucial tool for the study of the orbit structure of minimal ℤn-actions on the Cantor set, see Remark 4.8. The absorption theorem is a significant generalization of the main theorem proved in Giordano et al [Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys.24 (2004), 441–475] . However, we shall need a few key results from the above paper in order to prove the absorption theorem.

Let Γ be a countable group and let Γ0 be an infinite abelian subgroup of Γ. We prove that if the pair (Γ,Γ0) satisfies some combinatorial condition called (SS), then the abelian subalgebra A=L(Γ0) is a singular MASA in M=L(Γ) which satisfies a weakly mixing condition. If, moreover, it satisfies a stronger condition called (ST), then it provides a singular MASA with a strictly stronger mixing property. We describe families of examples of both types coming from free products, Higman–Neumann–Neumann extensions and semidirect products, and in particular we exhibit examples of singular MASAs that satisfy the weak mixing condition but not the strong mixing one.

Let (M,g) be a compact, smooth Riemannian manifold without conjugate points whose geodesic flow is expansive. We show that the geodesic flow of (M,g) has the accessibility property, namely, given two points θ1, θ2 in the unit tangent bundle, there exists a continuous path joining θ1, θ2 formed by the union of a finite number of continuous curves, each of which is contained either in a strong stable set or in a strong unstable set of the dynamics. Since expansive geodesic flows of compact surfaces have no conjugate points, the accessibility property holds for every two-dimensional expansive geodesic flow.

Working with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamic formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved.

In this article we prove an analogue of Bauer’s theorem from algebraic number theory in the context of hyperbolic systems.

It is well known that a generic polynomial foliation of ℂ2 is minimal and ergodic. In this paper we prove an analogous result for analytic foliations.

We study stability questions for dynamics on using holomorphic families, which we require to be locally simply parameterized. Given such a holomorphic family , we define the notion of a ‘Fatou section’, which is intuitively a choice of one point in for each map in the family such that: (1) the chosen point depends holomorphically on M, and (2) the points chosen for two different maps behave dynamically ‘comparably’ under iteration of those respective maps. We prove a weak version of the λ-lemma for dynamical systems on by showing that the set of all Fatou sections is a compact space. We introduce the notion of a postcritically bounded holomorphic family and show that if a (locally simply parameterized) holomorphic family is postcritically bounded by some open subset of then: (1) repelling periodic points in the Julia set for one member of the family cannot bifurcate, nor can they leave the Julia set in other members of the family, (2) there is a Fatou section which never leaves the Julia set through each point of the Julia set of any member of the family, and (3) any intersection between the Julia set and the critical set is persistent, in the sense that there is a Fatou section through the point of intersection which never leaves either the critical set or the Julia set. Here, for conciseness, we are using the term ‘Julia set’ to mean the support of the unique measure of maximal entropy, although we recognize that how this term should be used in higher dimensions is still undecided. We also provide a brief summary of the development of the field in our introduction.

We study an extension of the Viana map where the base dynamics is a discontinuous expanding map, and prove the existence of two positive Lyapunov exponents.

We consider analytic Hamiltonian systems with two degrees of freedom, and prove that every Hamiltonian on the strong local stable manifold of the renormalization group fixed point obtained in Koch [A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. A 11 (2004), 881–909] has a non-differentiable golden invariant torus (conjugacy to a linear flow).

Let A be a unital simple -algebra of real rank zero. Given an isomorphismγ1:K1(A)→K1(A), we show that there is an automorphism α:A→A such that α*1=γ1 and α has the tracial Rokhlin property. Consequently, the crossed product is a simple unital AH-algebra with real rank zero. We also show that automorphisms with the Rokhlin property can be constructed from minimal homeomorphisms on a connected compact metric space.