In this paper we extend results concerning conservativity and the existence of σ-finite measures to random transformations which admit a countable relative Markov partition. We consider random systems which are locally fibre-preserving and which admit a countable, relative Markov partition. If the system is relative irreducible and satisfies a relative distortion property we deduce that the system is either totally dissipative or conservative and ergodic. For conservative systems, we provide sufficient conditions for the existence of absolutely continuous σ-finite invariant measures.

We propose a new model of ergodic optimization for expanding dynamical systems: the holonomic setting. In fact, we introduce an extension of the standard model used in this theory. The formulation we consider here is quite natural if one wants a meaning for possible variations of a real trajectory under the forward shift. In other contexts (for twist maps, for instance), this property appears in a crucial way. A version of the Aubry–Mather theory for symbolic dynamics is introduced. We are mainly interested here in problems related to the properties of maximizing probabilities for the two-sided shift. Under the transitive hypothesis, we show the existence of sub-actions for Hölder potentials also in the holonomic setting. We analyze then connections between calibrated sub-actions and the Mañé potential. A representation formula for calibrated sub-actions is presented, which drives us naturally to a classification theorem for these sub-actions. We also investigate properties of the support of maximizing probabilities.

We study a particle moving at unit speed in a channel made by connected self-similar billiard tables that grow in size by a factor r>1 from left to right (this model was recently introduced in the physics literature by Barra, Gilbert and Romo). Let q(T) denote the position of the particle at time T. Our main result is the existence of an asymptotic distribution of q(T)/T as and {ln T/ln r}→ρ for some 0≤ρ<1.

We consider the quadratic family of maps given by fa(x)=1−ax2 with x∈[−1,1], where a is a Benedicks–Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes X0,X1,… , given by Xn=fan, for every integer n≥0, where each random variable Xn is distributed according to the unique absolutely continuous, invariant probability of fa. Using techniques developed by Benedicks and Carleson, we show that the limiting distribution of Mn=max {X0,…,Xn−1} is the same as that which would apply if the sequence X0,X1,… was independent and identically distributed. This result allows us to conclude that the asymptotic distribution of Mn is of type III (Weibull).

A new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable group G, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any , an uncountable family of cpe actions of entropy h, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that if αG is co-induced from an action αΓ of a subgroup Γ, then h(αG)=h(αΓ). We also prove that if αΓ is a non-Bernoulli cpe action of Γ, then αG is also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of , which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

We introduce and study the notion of weak product recurrence. Two sufficient conditions for this type of recurrence are established. We deduce that any point with a dense orbit in either the full one-sided shift on a finite number of symbols or a mixing subshift of finite type is weakly product recurrent. This observation implies that distality does not follow from weak product recurrence. We have therefore answered, in the negative, a question posed by Auslander and Furstenberg.

A linear map from one Euclidean space to another may map a compact set bijectively to a set of smaller Hausdorff dimension. For ‘homogeneous’ fractals (to be defined), there is a phenomenon of ‘dimension conservation’. In proving this we shall introduce dynamical systems whose states represent compactly supported measures in which progression in time corresponds to progressively increasing magnification. Application of the ergodic theorem will show that, generically, dimension conservation is valid. This ‘almost everywhere’ result implies a non-probabilistic statement for homogeneous fractals.

A polynomial skew product of ℂ2 is a map of the form f(z,w)=(p(z),q(z,w)), where p and q are polynomials, such that f extends holomorphically to an endomorphism of ℙ2 of degree at least two. For polynomial maps of ℂ, hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson [Dynamics of polynomial skew products on C2. Math. Ann.314(3) (1999), 403–447] established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present an analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors.

We consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of Lp spaces. This contrasts with the positive behavior exhibited by these averages in a different range, as proved in Demeter et al [Maximal multilinear operators. Trans. Amer. Math. Soc.360(9) (2008), 4989–5042]. We also prove that the trilinear Hilbert transform is unbounded in a similar range of Lp spaces. The principle underlying these constructions is stated, setting the stage for more general results.

A germ of a singular foliation in is built, with its analytical class of separatrix and holonomy representations prescribed. Thanks to this construction, we study the link between the moduli space of a foliation and the moduli space of its separatrix.

We consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

We prove that every one-parameter family of area-preserving maps unfolding a homoclinic tangency has a sequence of parameter intervals, approaching the bifurcation parameter, where the dynamics exhibits wild hyperbolic sets accumulated by elliptic isles. This is a parametric conservative analogue of a famous theorem of Newhouse on the abundance of wild hyperbolic sets.

In this paper we prove the equivalence of two definitions of laminated currents.

In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [P. Walters. Trans. Amer. Math. Soc.353(1) (2001), 327–347], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in our earlier work [A. Johansson and A. Öberg. Math. Res. Lett.10(5–6) (2003), 587–601]. We also prove uniqueness of the so-called G-measures, introduced by Brown and Dooley [G. Brown and A. H. Dooley. Ergod. Th. & Dynam. Sys.11 (1991), 279–307], under square summability of variations.

We consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on , and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent α∈ ]0,2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators Pt,Tt. The operator Pt is a Fourier operator and is considered here as a perturbation of the Markov operator P=P0 of the random walk. The operator Tt is related to Pt by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T0=T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T.

We study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner–King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing and minimal self joinings. The latter two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.

Let be a compact connected two-dimensional manifold, with or without boundary, and let be a continuous map. We prove that if is a minimal set of the dynamical system then either or M is a nowhere dense subset of . Moreover, we add a shorter proof of the recent result of Blokh, Oversteegen and Tymchatyn, that in the former case is a torus or a Klein bottle.

Two results on textile systems are obtained. Using these we prove that for any automorphism φ of any topologically-transitive subshift of finite type, if φ is expansive and φ or φ−1 has memory zero or anticipation zero, then φ is topologically conjugate to a subshift of finite type. Moreover, this is generalized to a result on chain recurrent onto endomorphisms of topologically-transitive subshifts of finite type. Using textile systems and textile subsystems, we develop a structure theory concerning expansiveness with the pseudo orbit tracing property on directionally essentially weakly one-sided resolving automorphisms and endomorphisms of subshifts.

It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (ℝ2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in ℂ2 is not available at the present time.

We re-investigate the theory of deformations of tilings using P-equivariant cohomology. In particular, we relate the notion of asymptotically negligible shape functions introduced by Clark and Sadun to weakly P-equivariant forms. We then investigate more closely the relation between deformations of patterns and homeomorphism or topological conjugacy of pattern spaces.