Let X be a separable infinite-dimensional Banach space, and T a bounded linear operator on X; T is hypercyclic if there is a vector x in X with dense orbit under the action of T. For a fixed ε∈(0,1), we say that T is ε-hypercyclic if there exists a vector x in X such that for every non-zero vector y∈X there exists an integer n with . The main result of this paper is a construction of a bounded linear operator T on the Banach space ℓ1 which is ε-hypercyclic without being hypercyclic. This answers a question from V. Müller [Three problems, Mini-Workshop: Hypercyclicity and linear chaos, organized by T. Bermudez, G. Godefroy, K.-G. Grosse-Erdmann and A. Peris. Oberwolfach Rep.3 (2006), 2227–2276].

Anderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which ‘forces its border’. One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson–Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

This paper introduces Markov chains and processes over non-abelian free groups and semigroups. We prove a formula for the f-invariant of a Markov chain over a free group in terms of transition matrices that parallels the classical formula for the entropy a Markov chain. Applications include free group analogues of the Abramov–Rohlin formula for skew-product actions and Yuzvinskii’s addition formula for algebraic actions.

We consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the ‘overlap set’ 𝒪 is finite, and which are ‘invertible’ on the attractor A, in the sense that there is a continuous surjection q:A→A whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that q is not a local homeomorphism precisely at 𝒪. We suppose also that there is a rational function p with the Julia set J such that (A,q) and (J,p) are conjugate. We prove that if A has bounded turning and p has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS {λz,λz+1} where λ is a complex parameter in the unit disk, such that its attractor Aλ is a dendrite, which happens whenever 𝒪 is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map qλ on Aλ. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map pc (z)=z2 +c, with the Julia set Jc such that (Aλ,qλ) and (Jc,pc) are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.

In this paper, we introduce a C*-algebra associated with a primitive substitution. We show that when σ is proper, the C*-algebra is simple and purely infinite and contains the associated Cuntz–Krieger algebra and the crossed product C*-algebra of the corresponding Cantor minimal system. We calculate the K-groups.

In this paper we prove that the Poincaré map associated to a Lorenz-like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τr(x,x0) is the time needed for the orbit of a point x to enter a ball Br(x0) centered at x0, with small radius r, for the first time. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the physical measure at x0: for each x0 such that the local dimension dμ (x0) exists, holds for μ almost each x. In a similar way, it is possible to consider a quantitative recurrence indicator quantifying the speed of an orbit to come back to its starting point. Similar results hold for this recurrence indicator.

We construct an infinite sequence of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of Abelian differentials. The existence of such a sequence shows that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy–Veech induction strategy.

We study the dynamics of a type of non-conservative billiards where the ball is ‘kicked’ by the wall giving a new impulse in the direction of the normal. For different types of billiard tables we study the existence of attractors with dominated splitting.

The purpose of this paper is to exhibit highly structured subdynamics for a class of non-expansive algebraic ℤd-actions based on the closed orbits of elements of an action. This is done using dynamical Dirichlet series to encode orbit counts. It is shown that there is a distinguished group homomorphism from ℤd onto a finite abelian group that controls the form of the Dirichlet series of elements of an action and that these series have common analytic properties. Corresponding orbit growth asymptotics are subsequently investigated.

Let K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

In this paper we deal with the existence of periodic orbits of geodesible vector fields on closed 3-manifolds. A vector field is geodesible if there exists a Riemannian metric on the ambient manifold making its orbits geodesics. In particular, Reeb vector fields and vector fields that admit a global section are geodesible. We will classify the closed 3-manifolds that admit aperiodic volume-preserving Cω geodesible vector fields, and prove the existence of periodic orbits for Cω geodesible vector fields (not volume preserving), when the 3-manifold is not a torus bundle over the circle. We will also prove the existence of periodic orbits of C2 geodesible vector fields on some closed 3-manifolds.

We consider cubic polynomials with a simple parabolic fixed point of multiplier 1. For those maps, we prove that the boundary of the immediate basin of attraction of the parabolic point is a Jordan curve (except for the polynomial z+z3 where it consists in two Jordan curves). Moreover, we give a description of the dynamics and obtain the local connectivity of the Julia set under some assumptions.

We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere ) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups G such that G is generated by a compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. In one of the classes, we have that, for almost every sequence , the Julia set Jγ of γ is a Jordan curve but not a quasicircle, the unbounded component of is a John domain, and the bounded component of is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups G such that the planar postcritical set of G is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.