We prove that a ‘positive probability’ subset of the boundary of ‘{uniformly expanding circle transformations}’ consists of Kupka–Smale maps. More precisely, we construct an open class of two-parameter families of circle maps $(f_{a,\theta})_{a,\theta}$ such that, for a positive Lebesgue measure subset of values of $a$, the family $(f_{a,\theta})_\theta$ crosses the boundary of the uniformly expanding domain at a map for which all periodic points are hyperbolic (expanding) and no critical point is pre-periodic. Furthermore, these maps admit an absolutely continuous invariant measure. We also provide information about the geometry of the boundary of the set of hyperbolic maps.

A dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of , or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

We prove that for any self-conformal measures, without any separation conditions, the multifractal formalism partially holds. The result follows by establishing certain Gibbs properties for self-conformal measures.

We study a skew product with a curve of neutral points. We show that there exists a unique absolutely continuous invariant probability measure, and that the Birkhoff averages of a sufficiently smooth observable converge to a normal law or a stable law, depending on the average of the observable along the neutral curve.

We introduce a new tool to study the spectral type of rank-one transformations using the method of the central limit theorem for trigonometric sums. We get some new applications.

The existence of non-continuous invariant graphs (or strange non-chaotic attractors) in quasiperiodically forced systems has generated great interest, but there are still very few rigorous results about the properties of these objects. In particular, it is not known whether the topological closure of such graphs is typically a filled-in set, i.e. consists of a single interval on every fibre, or not. We give a positive answer to this question for the class of so-called pinched skew products, where non-continuous invariant graphs occur generically, provided that the rotation number on the base is diophantine and the system satisfies some additional conditions. For typical parameter families these conditions translate to a lower bound on the parameter. On the other hand, we also construct examples where the non-continuous invariant graphs contain isolated points, such that their topological closure cannot be filled-in.

We consider a new orbit equivalence invariant for measure-preserving actions of groups on the probability space, $\sigma:G\rightarrow {\rm Aut}(X,\mu)$, denoted by $\chi_0(\sigma;G)$ and defined as the ‘intersection’ of the 1-cohomology group, H$^1(\sigma,G)$, with Connes' invariant, $\chi(M)$, of the cross product von Neumann algebra, $M=L^\infty(X,\mu)\rtimes_\sigma G$. We calculate $\chi_0(\sigma;G)$ for certain actions of groups of the form $G=H\times K$ with $H$ non-amenable and $K$ infinite amenable and we deduce that any such group has uncountably many orbit inequivalent actions.

A Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A dominated splitting is a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of an adapted metric for a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub. Invariant Manifolds(Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.

We investigate the convergence, in norm and almost everywhere (a.e.), of weighted ergodic averages as well as weighted sums of independent identically distributed (iid) random variables. The averages are true ones, normalized by the corresponding sums of weights, which are only assumed to be non-negative. The $L_2$-norm convergence in the mixing case is shown to rely upon very simple conditions on the weights. We show that ‘quasimonotone weights’ with a simple additional condition yield a.e. convergence of weighted averages for all Dunford–Schwartz contractions of probability spaces and $L_1$-functions. For independent random variables, we look at weighted averages of centered random variables with bounded variances (or bounded moments of some order greater than 1), in particular the iid case, and obtain several sufficient conditions on the weights for almost sure convergence (weighted SLLN). For example, in Theorem 4.14 we show that if a weight sequence $\{w_k\}$ with divergent partial sums $W_n$ satisfies

\[ \sup_{n\ge 1} \frac1{W_n} \sum_{k=1}^n w_k ({\rm log}(w_k+1))^\beta < \infty\quad \text{for some }\beta > 1 \]

then for any iid sequence in the class $L({\rm log}^+L)^{1+\epsilon}$ the weighted averages converge almost surely to the expectation.

We prove that if is a finitely generated abelian group of orientation preserving C1 diffeomorphisms of which leaves invariant a compact set then there is a common fixed point for all elements of . We also show that if is any abelian subgroup of orientation preserving C1 diffeomorphisms of S2 then there is a common fixed point for all elements of a subgroup of with index at most two.

One of the fundamental properties of the Mandelbrot set is that the set of postcritically finite parameters is structured like a tree. We extend this result to the set of quadratic kneading sequences and show that this space contains no irrational decorations. Along the way, we prove a combinatorial analogue to the correspondence principle of dynamic and parameter rays. Our key tool is to work simultaneously with the two equivalent combinatorial concepts of Hubbard trees and kneading sequences.

We present the quadratic Weyl sums with θ,x∈[0,1) as cocycles over a measure-preserving transformation on the two-dimensional torus. We show then that these cocycles are not coboundaries for every irrational θ∈[0,1), and that for a dense Gδ set of θ∈[0,1) the corresponding skew product is ergodic. For each of those θ, there exists a dense Gδ set of full measure of x∈[0,1) for which the sequence , n=1,2,… , is dense in .

In this paper we prove that there exist bounded orbit injections from minimal $\mathbb{Z}^{2}$ actions of a Cantor set $T$ and $S$ into a common action $R$ if and only if the suspension spaces associated to $T$ and $S$ are homeomorphic. In this way we prove a two-dimensional analog of a result of Parry and Sullivan on flow equivalence and discrete cross-sections for minimal systems. At the same time the result is a topological analog of a result of del Junco and Rudolph on Kakutani equivalence for ergodic $\mathbb{Z}^{d}$ actions. We also prove a structural result about such suspension spaces. Namely, that they are a finite union of products of Cantor sets with polygons, $C_{i}\times P_{i}$, after an identification on the boundary, $C_{i}\times \partial P_{i}$, with the action given by ${\mathbb{R}}^{2}$ on the polygon. The polygons $P_{i}$ can be chosen to have properties associated with Voronoi or Delaunay tilings corresponding to a set of points located uniformly throughout the plane.

Let $M$ be a finite von Neumann algebra with the Haagerup property, and let $G$ be a compact group that acts continuously on $M$ and that preserves some finite trace $\tau$. We prove that, if $\Gamma$ is a countable subgroup of $G$ which has the Haagerup property, then the crossed product algebra $M\rtimes\Gamma$ also has the Haagerup property. In particular, we study some ergodic, non-weakly mixing actions of groups with the Haagerup property on finite, injective von Neumann algebras, and we prove that the associated crossed product von Neumann algebras are $\textrm{II}_1$-factors with the Haagerup property. Moreover, if the actions have property $(\tau)$, then the latter factors are full.

In this paper we complete the analysis of the stability of periodic points of piecewise isometric systems that are defined in the entire Euclidean space. We focus on the case where the determinant of the matrix ${\rm Id}-R_{w}$ is zero, where $R_{w}$ is the linear part of the return map generated by the periodic point whose coding is made of a countable repetition of the block $w$. The case where the determinant of ${\rm Id}-R_{w}$ is non-zero was studied in Mendes and Nicol (Periodicity and recurrence in piecewise rotations of Euclidean spaces. Int. J. Bif. Chaos, 14(7), 2353–2361) but still the statement is included here for the sake of completeness.

Let T be an orientation-preserving Lipschitz expanding map of the circle . A pre-image selector is a map with finitely many discontinuities, each of which is a jump discontinuity, and such that τ(x)∈T−1(x) for all . The closure of the image of a pre-image selector is called a flower and a flower with p connected components is called a p-flower. We say that a real-valued Lipschitz function can be Lipschitz flattened on a flower whenever it is Lipschitz cohomologous to a constant on that flower. The space of Lipschitz functions which can be flattened on a given p-flower is shown to be of codimension p in the space of all Lipschitz functions, and the linear constraints determining this subspace are derived explicitly. If a Lipschitz function f has a maximizing measure S which is Sturmian (i.e. is carried by a 1-flower), it is shown that f can be Lipschitz flattened on some 1-flower carrying S.

We prove that for $k\ge 5$ there does not exist a continuous map $\partial CV(F_k)\to\mathbb P\mathit{Curr}(F_k)$ that is either $\mathit{Out}(F_k)$-equivariant or $\mathit{Out}(F_k)$-anti-equivariant. Here $\partial CV(F_k)$ is the ‘length function’ boundary of Culler–Vogtmann's Outer space $CV(F_k)$, and $\mathbb P\mathit{Curr}(F_k)$ is the space of projectivized geodesic currents for $F_{k}$. We also prove that, if $k\ge 3$, for the action of $\mathit{Out}(F_k)$ on $\mathbb P\mathit{Curr}(F_{k})$ and for the diagonal action of $\mathit{Out}(F_k)$ on the product space $\partial CV(F_k)\times \mathbb P\mathit{Curr}(F_k)$, there exist unique non-empty minimal closed $\mathit{Out}(F_k)$-invariant sets. Our results imply that for $k\ge 3$ any continuous $\mathit{Out}(F_k)$-equivariant embedding of $CV(F_k)$ into $\mathbb P\mathit{Curr}(F_k)$ (such as the Patterson–Sullivan embedding) produces a new compactification of Outer space, different from the usual ‘length function” compactification $\overline{CV(F_k)}=CV(F_k)\cup \partial CV(F_k)$.

We show relations between new notions on cellular automata based on probabilistic dynamics: almost everywhere sensitivity to initial conditions for Besicovitch pseudo-distance, damage spreading (which measures the information (or damage) propagation), and the destruction of the initial configuration information. Through natural examples, we illustrate the relations between these formal definitions.

We find an explicit formula for the invariant density of a generalized β-map. This allows us to also find an explicit formula for the invariant density of Chebyshev map and discuss the monotonicity of the asymptotic average for such maps. Our results are based on a generalization of works of Parry (Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416; 15 (1964), 95–105).

It is known that the closure ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ of the orbit ${\mathop{\hbox{\rm Orb}}\nolimits}_{g}(x)$ of a point $x$ of a compact nilmanifold $X$ under a polynomial sequence $g$ of translations of $X$ is a disjoint finite union of sub-nilmanifolds of $X$. Assume that $g(0)=1_{G}$ and let $A$ be the group generated by the elements of $g$; we show in this paper that for almost all points $x\in X$, the closures ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ are congruent (that is, are translates of each other), with connected components of ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{g}(x)$ equal to (some of) the connected components of ${\mathop{\overline{\hbox{\rm Orb}}}\nolimits}_{A}(x)$.