Let f:M→ℝ be a Morse–Bott function on a compact smooth finite-dimensional manifold M. The polynomial Morse inequalities and an explicit perturbation of f defined using Morse functions fj on the critical submanifolds Cj of f show immediately that MBt(f)=Pt(M)+(1+t)R(t), where MBt(f) is the Morse–Bott polynomial of f and Pt(M) is the Poincaré polynomial of M. We prove that R(t) is a polynomial with non-negative integer coefficients by showing that the number of gradient flow lines of the perturbation of f between two critical points p,q∈Cj of relative index one coincides with the number of gradient flow lines between p and q of the Morse function fj. This leads to a relationship between the kernels of the Morse–Smale–Witten boundary operators associated to the Morse functions fj and the perturbation of f. This method works when M and all the critical submanifolds are oriented or when ℤ2 coefficients are used.

We provide effective versions of theorems of Furstenberg and Rudolph–Johnson regarding closed subsets and probability measures of ℝ/ℤ invariant under the action of a non-lacunary multiplicative semigroup of integers. In particular, we give an explicit rate at which the sequence {anbkx}n,k becomes dense for a,b fixed multiplicatively independent integers and x∈ℝ/ℤ Diophantine generic.

We study the dynamics of piecewise affine surface homeomorphisms from the point of view of their entropy. Under the assumption of positive topological entropy, we establish the existence of finitely many ergodic and invariant probability measures maximizing entropy and prove a multiplicative lower bound for the number of periodic points. This is intended as a step towards the understanding of surface diffeomorphisms. We proceed by building a jump transformation, using not first returns but carefully selected ‘good’ returns to dispense with Markov partitions. We control these good returns through some entropy and ergodic arguments.

This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.

We show that for T a Dunford–Schwartz operator on a σ-finite measure space (X,Σ,μ) and f∈L1(X,μ), whenever the one-sided ergodic Hilbert transform ∑ n≥1(Tnf/n) converges in norm, it converges μ-a.s. A similar result is obtained for any positive contraction of some fixed Lp(X,Σ,μ), p>1. Applying our result to the case where T is the (unitary) operator induced by a measure-preserving (invertible) transformation, we obtain a positive answer to a question of Gaposhkin.

We say that the geometric dimension of a countable group G is equal to n if any free Borel action of G on a standard Borel probability space (X,μ), induces an equivalence relation of geometric dimension n on (X,μ) in the sense of Gaboriau. Let ℬ be the set of all finitely generated amenable groups all of whose subgroups are also finitely generated, and let 𝒜 be the subset of ℬ consisting of finite groups, torsion-free groups and their finite extensions. In this paper we study finite free products K of groups in 𝒜. The geometric dimension of any such group K is one: we prove that also geom-dim(Gf(K))=1 for any finite extension Gf(K) of K, applying the results of Stallings on finite extensions of free product groups, together with the results of Gaboriau and others in orbit equivalence theory. Using results of Karrass, Pietrowski and Solitar we extend these results to finite extensions of free groups. We also give generalizations and applications of these results to groups with geometric dimension greater than one. We construct a family of finitely generated groups {Kn}n∈ℕ,n>1, such that geom-dim(Kn)=n and geom-dim(Gf(Kn))=n for any finite extension Gf(Kn) of Kn. In particular, this construction allows us to produce, for each integer n>1, a family of groups {K(s,n)}s∈ℕ of geometric dimension n, such that any finite extension of K(s,n) also has geometric dimension n, but the finite extensions Gf(K(s,n)) are non-isomorphic, if s≠s′.

For points in d real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed d by d matrix over ℤ. Our starting point is a given pair (A,𝒟) with the matrix A assumed expansive, and 𝒟 a chosen complete digit set, i.e., in bijective correspondence with the points in ℤd/ATℤd. We give an explicit geometric representation and encoding with infinite words in letters from 𝒟. We show that the attractor X(AT,𝒟) for an affine Iterated Function System (IFS) based on (A,𝒟) is a set of fractions for our digital representation of points in ℝd. Moreover our positional ‘number representation’ is spelled out in the form of an explicit IFS-encoding of a compact solenoid 𝒮A associated with the pair (A,𝒟). The intricate part (Theorem 6.15) is played by the cycles in ℤd for the initial (A,𝒟)-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional 𝒟-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces.

We study Birkhoff normalization in connection with superintegrability of an n-degree-of-freedom Hamiltonian system XH with holomorphic Hamiltonian H. Without assuming any Poisson commuting relation among integrals, we prove that, if the system XH has n+q holomorphic integrals near an equilibrium point of resonance degree q≥0, there exists a holomorphic Birkhoff transformation φ such that H∘φ becomes a holomorphic function of n−q variables and that XH∘φ can be solved explicitly. Furthermore, the Birkhoff normal form H∘φ is determined uniquely, independently of the choice of φ, as convergent power series. We also show that the system XH is superintegrable in the sense of Mischenko–Fomenko as well as Liouville-integrable near the equilibrium point.

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit circle, in various parameters, and applications to the study of quantum dynamics.

We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of Tao of the mean ergodic theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad et al [Local stability of ergodic averages. Trans. Amer. Math. Soc. to appear] and Tao [Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys.28(2) (2008), 657–688].

In this paper we consider horseshoes with homoclinic tangencies inside the limit set. For a class of such maps, we prove the existence of a unique equilibrium state μt, associated to the (non-continuous) potential −tlog Ju. We also prove that the Hausdorff dimension of the limit set, in any open piece of unstable manifold, is the unique number t0 such that the pressure of μt0 is zero. To deal with the discontinuity of the jacobian, we introduce a countable Markov partition adapted to the dynamics, and work with the first return map defined in a rectangle of it.

This article is about the interplay between topological dynamics and differential geometry. One could ask how much information about the geometry is carried in the dynamics of the geodesic flow. It was proved in Paternain [Expansive geodesic flows on surfaces. Ergod. Th. & Dynam. Sys.13 (1993), 153–165] that an expansive geodesic flow on a surface implies that there exist no conjugate points. Instead of considering concepts that relate to chaotic behavior (such as expansiveness), we focus on notions for describing the stability of orbits in dynamical systems, specifically, equicontinuity and distality. In this paper we give a new sufficient and necessary condition for a compact Riemannian surface to have all geodesics closed; this is the idea of a P-manifold: (M,g) is a P-manifold if and only if the geodesic flow SM×ℝ→SM is equicontinuous. We also prove a weaker theorem for flows on manifolds of dimension three. Finally, we discuss some properties of equicontinuous geodesic flows on non-compact surfaces and on higher-dimensional manifolds.

We consider C∞-diffeomorphisms on a Banach space with a fixed point 0 and linear part L. Suppose that these diffeomorphisms have C∞ non-contracting and non-expanding invariant manifolds, and formally conjugate along their intersection (the center). We prove that they admit local C∞ conjugation. In particular, subject to non-resonance conditions, there exists a local C∞ linearization of the diffeomorphisms. It also follows that a family of germs with a hyperbolic linear part admits a C∞ linearization, which has C∞ dependence on the parameter of the linearizing family. The results are proved under the assumption that the Banach space allows a special extension of the maps. We discuss corresponding properties of Banach spaces. The proofs of this paper are based on the technique, developed in the works of Belitskii [Funct. Anal. Appl.18 (1984), 238–239; Funct. Anal. Appl.8 (1974), 338–339].

We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number τ there exists a C∞ circle diffeomorphism with rotation number τ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value 0≤β≤1.