We characterize numbers having finite β-expansions where β belongs to a certain class of Pisot numbers: when the β-expansion of 1 is equal to a1a2…am, where a1≥a2≥…≥am≥1 and when the β-expansion of 1 is equal to t1t2…tm(tm+1)ω where t1≥t2≥…≥tm>tm+1≥1.

I use a thermodynamic formalism to study the spectrum f(α) which characterises the large fluctuations of pointwise dimension in a Gibbs state supported on a hyperbolic cookie-cutter. Amongst other things, it is proved thatf(α) is the Hausdorff dimension of the set of points with pointwise dimension α, that f(α) is real-analytic and that its Legendre transform τ(q) is related to the Renyi dimension Dq of the Gibbs state by the formula (1 − q)Dq = τ(q).

We show that, for any compact surface, there is a residual (dense G_{\delta}) set of C^{1} area-preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponents a.e. This result was announced by R. Mañé, but no proof was available. We also show that for any fixed ergodic dynamical system over a compact space, there is a residual set of continuous SL(2,\mathbb{R})-cocycles which either are uniformly hyperbolic or have zero exponents a.e.

We provide an algorithm to compute the first non-zero derivative of the return map r ↦ L(r, ε) of a planar vector field which is a polynomial perturbation of . It yields a new method for finding the centre conditions in the case of a homogeneous perturbative part.

We explore substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit and render an effective method to compute an invariant of (ordered) $K$-theoretic nature for these systems. This new invariant is independent of spectral invariants which have previously been extensively studied. Before we state the main results we give some background.

We give a proof of Pesin entropy formula in a very general setting.

The dynamical system associated to the difference equation

has been studied numerically by several authors. On the basis of numerical evidence, they conclude that there exists a number k0 ≈ 0.97 such that there are homotopically non-trivial invariant circles for |k|≤k0 and there are none for |k|>k0. In this note, we give a simple rigorous proof that there are none for |k|>.

More than 30 years ago R\'enyi [1] introduced the representations of real numbers with an arbitrary base $\beta > 1$ as a generalization of the $p$-adic representations. One of the most studied problems in this field is the link between expansions to base $\beta$ and ergodic properties of the corresponding $\beta$-shift.

In this paper we will follow the bibliography of Blanchard [2] and give an affirmative answer to a question on the size of the set of real numbers $\beta$ having complicated symbolic dynamics of their $\beta$-shifts.

Let $(X,d,T)$ be a dynamical system, where $(X,d)$ is a compact metric space and $T:X\rightarrow X$ a continuous map. We introduce two conditions for the set of orbits, called respectively the $\texttt{g}$-almost product property and the uniform separation property. The $\texttt{g}$-almost product property holds for dynamical systems with the specification property, but also for many others. For example, all $\beta$-shifts have the $\texttt{g}$-almost product property. The uniform separation property is true for expansive and more generally asymptotically $h$-expansive maps. Under these two conditions we compute the topological entropy of saturated sets. If the uniform separation condition does not hold, then we can compute the topological entropy of the set of generic points, and show that for any invariant probability measure $\mu$, the (metric) entropy of $\mu$ is equal to the topological entropy of generic points of $\mu$. We give an application of these results to multi-fractal analysis and compare our results with those of Takens and Verbitskiy (Ergod. Th. & Dynam. Sys.23 (2003), 317–348).

We show that in several cases preservation of cones leads to non-vanishing of (some) Lyapunov exponents. It gives simple and effective criteria for nonvanishing of the exponents, which is demonstrated on the example of the billiards studied by Bunimovich. It is also shown that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view.

Let T1,…,Tl:X→X be commuting measure-preserving transformations on a probability space . We show that the multiple ergodic averages are convergent in as for all ; this was previously established for l=2 by Conze and Lesigne [J. P. Conze and E. Lesigne. Théorèmes ergodique por les mesures diagonales. Bull. Soc. Math. France112 (1984), 143–175] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiTj−1 by Frantzikinakis and Kra [N. Frantzikinakis and B. Kra. Convergence of multiple ergodic averages for some commuting transformations. Ergod. Th. & Dynam. Sys.25 (2005), 799–809] (with the l=3 case of this result established earlier by Zhang [Q. Zhang. On the convergence of the averages . Mh. Math.122 (1996), 275–300]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l=2 case of our arguments is a finitary analogue of those by Conze and Lesigne.

We are interested in ergodic properties of absolutely continuous invariant measures of positive entropy for a map of an interval. We prove a Bernoulli property and a characterization by some variational principle.

Consider an analytic differential equation ο = adx + bdy with an algebraically isolated singularity and without a separatrix. The germ at 0∈ℝ2 of the 1-jet y dy is either a focus or a centre. The equation has C∞ normal form of the type = with F(x) = F(−x) if and only if the germ is a centre.

We describe the dynamical behaviour of the entire transcendental function exp(z). We use symbolic dynamics to describe the complicated orbit structure of this map whose Julia Set is the entire complex plane. Bifurcations occurring in the family c exp(z) are discussed in the final section.

Suppose that (X, ℬ, μ) is a probability measure space and T is an invertible measure perserving transformation of (X, ℬ, μ). T is called weakly mixing if for any two sets A1A2 ∈ ℬ one has:

We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This implies that the ergodic averages of a continuous function on X along a polynomial sequence of translations on X converge pointwise.

Certain topological dynamical systems that arise from actions of $\sigma$-compact locally compact Abelian groups on compact spaces of translation bounded measures are considered. Such a measure dynamical system is shown to have a pure point dynamical spectrum if and only if its diffraction spectrum is pure point.

We consider iterates of absolutely continuous measures concentrated in a neighbourhood of a partially hyperbolic attractor. It is shown that limit points can be measures which have conditional measures of a special form for any partition into subsets of unstable manifolds.

We study asymptotic growth of closed geodesies for various Riemannian metrics on a compact manifold which carries a metric of negative sectional curvature. Our approach makes use of both variational and dynamical description of geodesies and can be described as an asymptotic version of length-area method. We also obtain various inequalities between topological and measure-theoretic entropies of the geodesic flows for different metrics on the same manifold. Our method works especially well for any metric conformally equivalent to a metric of constant negative curvature. For a surface with negative Euler characteristics every Riemannian metric has this property due to a classical regularization theorem. This allows us to prove that every metric of non-constant curvature has strictly more close geodesies of length at most T for sufficiently large T then any metric of constant curvature of the same total area. In addition the common value of topological and measure-theoretic entropies for metrics of constant negative curvature with the fixed area separates the values of two entropies for other metrics with the same area.

In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.