In this paper we consider twist mappings of the torus, $\overline{T}:{\rm T^2\rightarrow T^2}$, and their vertical rotation intervals $\rho _V(T)=[\rho _V^{-},\rho _V^{+}]$, which are closed intervals such that for any $\omega \in\, ]\rho _V^{-},\rho _V^{+}[$ there exists a compact $\overline{T}$-invariant set $\overline{Q}_\omega $ with $\rho _V(\overline{x})=\omega$ for any $\overline{x}\in \overline{Q}_\omega $, where $\rho _V(\overline{x})$ is the vertical rotation number of $\overline{x}$. In the case when $\omega $ is a rational number, $\overline{Q}_\omega $ is a periodic orbit. Here we analyze how $\rho _V^{-}$ and $\rho _V^{+}$ behave as we perturb $\overline{T}$ and which dynamical properties for $\overline{T}$ can be obtained from their values.

We introduce the notion of an Anosov family, a generalization of an Anosov map of a manifold. This is a sequence of diffeomorphisms along compact Riemannian manifolds such that the tangent bundles split into expanding and contracting subspaces. We develop the general theory, studying sequences of maps up to a notion of isomorphism and with respect to an equivalence relation generated by two natural operations, gathering and dispersal. Then we concentrate on linear Anosov families on the 2-torus. We study in detail a basic class of examples, the multiplicative families, and a canonical dispersal of these, the additive families. These form a natural completion to the collection of all linear Anosov maps. A renormalization procedure constructs a sequence of Markov partitions consisting of two rectangles for a given additive family. This codes the family by the non-stationary subshift of finite type determined by exactly the same sequence of matrices. Any linear positive Anosov family on the torus has a dispersal which is an additive family. The additive coding then yields a combinatorial model for the linear family, by telescoping the additive Bratteli diagram. The resulting combinatorial space is then determined by the same sequence of non-negative matrices, as a non-stationary edge shift. This generalizes and provides a new proof for theorems of Adler and Manning.

Let X be a graph and let G be a subgroup of the automorphism group of X. We investigate the question of when the full 2-shift process on the vertices of X has a G-factor process on the same graph consisting of random variables which are independent and identically distributed (i.i.d.) uniform in [0, 1]. We show that such a factor always exists when X is a tree with bounded degrees, no leaves and at least three topological ends, and give a sufficient condition for when such a factor exists if X is a connected graph with infinitely many ends and G acts transitively on the vertices of X. We also show that the full 2m-shift on any finitely-generated non-abelian group G has a G-factor which is i.i.d. uniform on [0, 1] when m is sufficiently large.

We determine the Hausdorff and box dimensions of the flexed Sierpiński gasket $\Lambda$, which is the limit set for a non-Moran-like non-affine geometric construction in the plane. We also provide a multifractal formalism for the Bernoulli measures on $\Lambda$, computing explicitly the pointwise dimension spectra.

This work is devoted to the study of periodic solutions of generalized Lotka–Volterra systems in three dimensions. The group A3 of isometries of the regular tetrahedron acts on the twelve-dimensional space of such systems. We find four components (modulo the action of A3) of the center variety, i.e. the semi-algebraic subset consisting of systems with a family of periodic solutions (which degenerate at the center). We focus attention on the component which consists of systems X0 with an invariant plane L0 transversal to the coordinate axes and with a (partial) first integral on L0. Under the assumption of normal hyperbolicity of X0 along L0 we begin study of limit cycles appearing for a perturbed system $X_{0}+\varepsilon X_{1}$ (in the Lotka–Volterra class). The linearization of the problem leads to the problem of zeroes of certain Poincaré–Melnikov integrals of a new type. We determine the asymptotic behavior of these integrals near two critical values of the partial first integral of the restricted system X0|L0. We also study the situation near the intersection of different components of the center variety.

We define the concept of horospherical points for flows admitting a local product structure, and show that these points have a dense strong stable set. This generalizes a classical result from G. A. Hedlund, in the context of geodesic flows on negatively curved manifolds.

We prove the L2-convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, \dots, T_{l}$, assuming that each map Ti and each pair TiTj-1 is ergodic for $i\neq j$. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.

A one-dimensional shift of finite type $(X, \mathbb Z)$ with entropy at least log n factors onto the full n-shift. The factor map is constructed by exploiting the fact that X, or a subshift of X, is conjugate to a shift of finite type in which every symbol can be followed by at least n symbols. We will investigate analogous statements for higher-dimensional shifts of finite type. We will also show that for a certain class of mixing higher-dimensional shifts of finite type, sufficient entropy implies that $(X,\mathbb Z^d )$ is finitely equivalent to a shift of finite type that maps onto the full n-shift.

The two-sided shift on the infinite tensor product of copies of the $n\times n$ matrix algebra has the so-called Rohlin property, which entails the one-cocycle property, useful in analyzing cocycle-conjugacy classes. In the case n = 2, the restriction of the shift to the gauge-invariant CAR algebra also has the one-cocycle property. We extend the latter result to an arbitrary $n\geq 2$. As a corollary it follows that the flow $\alpha$ on the Cuntz algebra $\mathcal{O}_n=C^*(s_0,s_1,\dotsc,s_{n-1})$ defined by $\alpha_t(s_j)=e^{ip_jt}s_j$ has the Rohlin property (for flows) if and only if $p_0,\dotsc,p_{n-1}$ generate $\mathbb R$ as a closed sub-semigroup. Note that then such flows are all cocycle-conjugate to each other.

We present several results for the periods of periodic points of sup-norm non-expansive maps. In particular, we show that the period of each periodic point of a sup-norm non-expansive map $f\colon M\to M$, where $M\subset \mathbb{R}^n$, is at most $\max_k\, 2^k \big(\begin{smallmatrix}n\\ k\end{smallmatrix}\big)$. This upper bound is smaller than 3n and improves the previously known bounds. Further, we consider a special class of sup-norm non-expansive maps, namely topical functions. For topical functions $f\colon\mathbb{R}^n\to\mathbb{R}^n$ Gunawardena and Sparrow have conjectured that the optimal upper bound for the periods of periodic points is $\big(\begin{smallmatrix}n\\ \lfloor n/2\rfloor\end{smallmatrix}\big)$. We give a proof of this conjecture. To obtain the results we use combinatorial and geometric arguments. In particular, we analyse the cardinality of anti-chains in certain partially ordered sets.

We build a Markov partition of the Feigenbaum dynamical plane, and a system of ‘external rays’, which are invariant under rescaling. Applications to geometry of the Feigenbaum map are given.

In this work, singular orientable foliations which admit non-trivial recurrent leaves on 2-manifolds of finite or infinite genus are considered. A structure theorem for such foliations is given. The theorem is similar to Gutierrez's structure theorem (C. Gutierrez. Smoothing continuous flows on 2-manifolds and recurrences. Ergod. Th. & Dynam. Sys.6 (1986), 17–44) for flows on compact surfaces.

We will give a sufficient condition for points in a locally defined analytic subvariety V in $\mathbb{P}^k$ to be contained in the Fatou set in V for a birational self-map of degree ${\ge} 2$ of $\mathbb{P}^k$. We will consider its applications.

Consider the three-body problem with an attractive 1/r2 potential. Modulo symmetries, the dynamics of the bounded zero-angular-momentum solutions is equivalent to a geodesic flow on the thrice-punctured sphere, or ‘pair of pants’. The sphere is the shape sphere. The punctures are the binary collisions. The metric generating the geodesics is the Jacobi–Maupertuis metric. It is complete, has infinite area, and its ends, the neighborhoods of the punctures, are asymptotically cylindrical. Our main result is that when the three masses are equal then the metric has negative curvature everywhere except at two points (the Lagrange points). A corollary of this negativity is the uniqueness of the 1/r2 figure-eight, a complete symbolic dynamics for encoding the collision-free solutions, and the fact that collision solutions are dense within the bound solutions.

Given a Lagrangian L, Mather introduced the $\beta$-function of L, which is a convex function. Many interesting properties of the Euler–Lagrange flow can be derived from the study of the behaviour of the $\beta$-function. In this work we obtain some links between the vertices of the $\beta$-function and the Hausdorff dimension of the associated invariant measures, from which we recover a result of differentiability of the $\beta$-function.