In this article, we consider semigroups of transformations of cellular automata which act on a fixed shift space. In particular, we are interested in two properties of these semigroups which relate to ‘largeness’: first, a semigroup has the ID (infinite is dense) property if the only infinite invariant closed set (with respect to the semigroup action) is the entire space; the second property is maximal commutativity (MC). We shall consider two examples of semigroups: one is spanned by cellular automata transformations that represent multiplications by integers on the one-dimensional torus, and the other one consists of all the cellular automata transformations which are linear (when the symbols set is the ring ℤ/sℤ). It will be shown that these two properties of these semigroups depend on the number of symbols s. The multiplication semigroup is ID and MC if and only if s is not a power of a prime. The linear semigroup over the mentioned ring is always MC but is ID if and only if s is prime. When the symbol set is endowed with a finite field structure (when possible), the linear semigroup is both ID and MC. In addition, we associate with each semigroup which acts on a one-sided shift space a semigroup acting on a two-sided shift space, and vice versa, in a way that preserves the ID and the MC properties.

In this paper we study the asymptotic behaviour of the escape rate of a Gibbs measure supported on a conformal repeller through a small hole. There are additional applications to the convergence of the Hausdorff dimension of the survivor set.

We discuss the dynamics of skew product maps defined by circle diffeomorphisms forced by expanding circle maps. We construct an open class of such systems that are robustly topologically mixing and for which almost all points in the same fiber converge under iteration. This property follows from the construction of an invariant attracting graph in the natural extension, a skew product of circle diffeomorphisms forced by a solenoid homeomorphism.

In this paper we present a surprising example of a Cr unimodal map of an interval f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’ of critical points is necessary for the Martens–de Melo–van Strien theorem [M. Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.

In this paper, we introduce and study the α-Farey map and its associated jump transformation, the α-Lüroth map, for an arbitrary countable partition α of the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called α-sum-level sets for the α-Lüroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the α-Farey map and the α-Lüroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition α.

We analyse dynamical properties of the negative beta-transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta-transformation, the density of the absolutely continuous invariant measure of the negative beta-transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (−β)-transformation is exact for all β>1, confirming a conjecture of Góra, and intrinsic, which completes a study of Faller. We also show that the limit behaviour of the (−β)-expansion of 1 when β tends to 1 is related to the Thue–Morse sequence. A consequence of the exactness is that every Yrrap number, which is a β>1 such that the (−β) -expansion of 1 is eventually periodic, is a Perron number. This extends a well-known property of Parry numbers. However, the set of Parry numbers is different from the set of Yrrap numbers.

We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular, we show that a closed subgroup of a product of finitely many linear groups over local fields supports an adapted recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces and for associated homeomorphisms with infinite invariant measure. The structural properties of closed subgroups of linear groups over local fields and the properties of group actions with respect to certain Radon measures associated with random walks play an important role in the proofs.

We characterize representations of finitely generated discrete groups into (the connected component of) the isometry group of a complex hyperbolic space via the pullback of the bounded Kähler class.

We prove that there is no minimal action of ℤ2 by homeomorphisms on the plane. This may be seen as a generalization of Le Calvez–Yoccoz’s theorem: there exists no minimal homeomorphism of the infinite annulus.

We extend the idea of bilateral determinism of a free Z-action by D. Ornstein and B. Weiss to a free Z2-action. We show that we have a ‘stronger’ spatial determinism for Z2-actions: to determine the complete Z2-name of a point, it is enough to know the name of a fraction of the orbit whose density can be made arbitrarily small. Moreover, for zero-entropy Z2-actions, we prove that there exists a partition such that the -names of an arbitrarily small one-sided cone determine the points.

We formulate and prove a precise statement of asymptotic shrinking of ‘carrot-fields’ around the Mandelbrot set M. This phenomenon was suggested in a seminal text on polynomial-like mappings by Douady and Hubbard [On the dynamics of polynomial-like mappings. Ann. Sci. Éc. Norm. Supér. t.18 (1985), 287–343]. This is helpful for understanding how copies of M sit in the bifurcation loci of families of rational maps.

In this paper, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational and transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Epstein. In fact, we prove a more general result on the absence of invariant differentials, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equations for log ∣f′∣, which are relevant to a number of well-known rigidity questions. In particular, we prove the absence of continuous line fields on the Julia set of any transcendental entire function.

We consider N-body problems with potential 1/r2κ, where κ∈(0,1), including the Newtonian case (κ=1/2). Given R>0 and T>0, we find a uniform upper bound for the minimal action of paths binding, in time T, any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates the Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding two configurations in free time). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax–Oleinik semigroup, and we show that they are global viscosity solutions of the corresponding Hamilton–Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

The derivation of Ruelle’s derivative formula of the SRB measure depends largely on the calculation of the derivative of the unstable Jacobian. Although Ruelle’s derivative formula is correct, the proofs in the original paper and its corrigendum are not complete. In this paper, we re-visit the differentiation process of the unstable Jacobian and provide a complete derivation of its derivative formula. Our approach is to extend the volume form provided by the SRB measure on local unstable manifolds to a system of Hölder continuous local Riemannian metrics on the manifold so that under this system of local metrics, the unstable Jacobian becomes differentiable with respect to the base point and its derivative with respect to the map can be obtained by the chain rule.

In this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

We study billiard dynamics on non-compact polygonal surfaces with a free, cocompact action of ℤ or ℤ2. In the ℤ-periodic case, we establish criteria for conservativity. In the ℤ2-periodic case, we study a particular family of such surfaces, the rectangular Lorenz gas. Assuming that the obstacles are sufficiently small, we obtain the ergodic decomposition of directional billiards for a finite but asymptotically dense set of directions. This is based on our study of ergodicity for ℤd-valued cocycles over irrational rotations.

Let Γ be a topological semigroup acting on a topological space X, and let Γ0 be a subsemigroup of Γ. We give general conditions ensuring that Γ and Γ0 have the same transitive points.

We investigate the distribution of orbits of a non-elementary discrete hyperbolic subgroup Γ acting on ℍn and its geometric boundary ∂∞(ℍn). In particular, we show that if Γ admits a finite Bowen–Margulis–Sullivan measure (for instance, if Γ is geometrically finite), then every Γ-orbit in ∂∞(ℍn) is equidistributed with respect to the Patterson–Sullivan measure supported on the limit set Λ(Γ). The appendix by Maucourant is the extension of a part of his PhD thesis where he obtains the same result as a simple application of Roblin’s theorem. Our approach is via establishing the equidistribution of solvable flows on the unit tangent bundle of Γ∖ℍn, which is of independent interest.

We study invariants for shifts of finite type obtained as the K-theory of various C*-algebras associated with them. These invariants have been studied intensively over the past thirty years since their introduction by Wolfgang Krieger. They may be given quite concrete descriptions as inductive limits of simplicially ordered free abelian groups. Shifts of finite type are special cases of Smale spaces and, in earlier work, the second author has shown that the hyperbolic structure of the dynamics in a Smale space induces natural ring and module structures on certain of these K-groups. Here, we restrict our attention to the special case of shifts of finite type and obtain explicit descriptions in terms of the inductive limits.

We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibered holomorphic maps. As in the classical non-fibered case, we show that petals are the main ingredient. Nevertheless, one expects that the properties of the base rotation number should play an important role in the arrangement of the petals. We exhibit examples where the existence and the number of petals depend not just on the complex coordinate of the map, but on the base rotation number. Furthermore, under additional hypotheses on the arithmetic and smoothness of the map, we present a theorem that allows a characterization of the local dynamics around a parabolic invariant curve.