Let $\mathcal{H}$ be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that $\mathcal{H}$ has compact closure in moduli space if and only if neither attractor is a fixed point.

We characterize when C0-semigroups induced by semiflows are hypercyclic, topologically mixing, or chaotic both on spaces of integrable functions and on spaces of continuous functions. Furthermore, we give characterizations of transitivity for weighted composition operators on these spaces.

We prove some results related to the question of the existence of factor maps and eventual factor maps between shifts of finite type. Our main result is that if A and B are integral eventually positive (IEP) matrices, and A eventually factors finite-to-one onto B, then there exists an IEP matrix C such that A eventually factors onto C by left closing maps and C eventually factors onto B by right closing maps. This answers the question of the existence of finite-to-one eventual factor maps when the spectrum of A is simple. As a corollary, if in addition to the above hypothesis, χ*A=χ*B, (where χ*A is the characteristic polynomial of A modulo x), then A is shift equivalent to B.

It is proved that an arbitrary one dimensional dynamical system with negative Schwarzian derivative and non-degenerate critical points has no wandering intervals. This result implies a rather complete view of the dynamics of such a system. In particular, every minimal topological attractor is either a limit cycle, or a one dimensional manifold with boundary, or a solenoid. The orbit of a generic point tends to some minimal attractor.

Let $\{h_{a,b} : a,b \in \mathbb{R}, b \neq 0\}$ denote the Hénon family of quadratic polynomial diffeomorphisms of $\mathbb{R}^2$, with $b$ equal to the Jacobian of $h_{a,b}$. In this paper, we describe the locus of parameter values $(a,b)$ such that $0 < |b| < 0.06$, and the restriction of $h_{a,b}$ to its non-wandering set is topologically conjugate to the horseshoe map. The boundary of the horseshoe locus is shown to be characterized by a homoclinic tangency which is part of a generic unfolding.

We show that the $n$-point correlation function for the fractional parts of a random linear form in $m$ variables has a limit distribution with power-like tail. The existence of the limit distribution follows from the mixing property of flows on ${\rm SL}(m+1,{\Bbb R})/{\rm SL}(m+1,{\Bbb Z})$. Moreover, we prove similar limit theorems (i) for the probability to find the fractional part of a random linear form close to zero and (ii) also for related trigonometric sums. For large $m$, all of the above limit distributions approach the classical distributions for independent uniformly distributed random variables.

We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.

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.

The Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dotsc over a finite alphabet, the authors introduced a new notion of complexity called maximal pattern complexity defined by p_\alpha^*(k):=\sup_\tau\sharp\{\alpha_{n+\tau(0)}\alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)};n=0,1,2,\dotsc\} where the supremum is taken over all sequences of integers 0=\tau(0)<\tau(1)<\dotsc<\tau(k-1) of length k. The authors proved that \alpha is aperiodic if and only if p_\alpha^*(k)\ge 2k for every k=1,2,\dotsc. A word \alpha with p_\alpha^*(k)=2k for every k\geq 1 is called pattern Sturmian. In this paper, we give a simple criterion to be pattern Sturmian and exhibit a new class of recurrent pattern Sturmian words which do not arise from rotations. We also investigate the maximal pattern complexity of various discrete dynamical systems including irrational rotations on the circle, and self-similar systems generated by substitutions. We show that, for each irrational rotation on the circle, there exists a twofold partition of the circle, with respect to which the system generated has full maximal pattern complexity with probability one. Using the arithmetic properties of the underlying numeration system associated to a substitution dynamical system, we prove that the maximal pattern complexity of the fixed point of the Rauzy substitution 1\mapsto 12, 2\mapsto 13, 3\mapsto 1 has exponential growth. It is well known that the system generated by the Rauzy substitution is isomorphic in measure to an irrational rotation on the 2-torus.

We combine recurrence properties of polynomials and IP-sets and show that polynomials evaluated along IP-sequences also give rise to Poincaré sets for measure-preserving systems, that is, sets of integers along which the analogue of the Poincaré recurrence theorem holds. This is done by applying to measure-preserving transformations a limit theorem for products of appropriate powers of a commuting family of unitary operators.

This paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of:

• • either a robust local phenomenon;

• • or a global structure forbidding this local phenomenon.

In the present paper, we introduce a natural extension of Adler, Konheim and MacAndrew topological entropy for proper maps of locally compact separable metrizable spaces and prove a variational principle which states that this topological entropy, the supremum of the Kolmogorov–Sinai entropies and the minimum of the Bowen entropies always coincide. We apply this variational principle to show that the topological entropy of automorphisms of simply connected nilpotent Lie groups always vanishes. This shows that the Bowen formula for the Bowen entropy of an automorphism of a non-compact Lie group with respect to some invariant metric is just an upper bound for its topological entropy.

The Lyapunov exponents of locally constant $\text{GL}(2,\mathbb{C})$-cocycles over Bernoulli shifts vary continuously with the cocycle and the invariant probability measure.

We study dissipative twist maps of the annulus, following the ideas of G. D. Birkhoff explained in an article of 1932.

In the first part, we give complete and rigorous proofs of the results of this article. We define the Birkhoff attractor of a dissipative twist map which has an attracting bounded annulus, we give its main properties and we define its upper and lower rotation numbers.

In the second part we give further results on these sets, thus we show that they often coincide with the closure of a hyperbolic periodic point and that they can contain an infinite number of sinks. We also show that the Birkhoff attractors don't depend on a continuous way on the maps.

We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex $S^{m-1}$ . We prove that in contrast to the deterministic set-up the trajectories of the random dynamical system almost surely converge to one of the vertices of the simplex $S^{m-1}$ , implying the survival of only one species. We also show that the minimal random point attractor of the system equals the set of all vertices. The convergence proof relies on a martingale-type limit theorem, which we prove in the appendix.

Let (X, T) be a topological dynamical system and let $\mu$ be a T-invariant probability measure on X. In this paper, we study two properties of the notions of measure theoretical entropy for a measurable cover $\mathcal{U},\ h_\mu^+(\mathcal{U},T)$ and $h_\mu^-(\mathcal{U},T)$ introduced by P. P. Romagnoli (Ergod. Th. & Dynam. Sys. 23 (2003), 1601–1610). The main result of the paper states that entropy pairs for the measure $\mu$ can be defined using either $h_\mu^+$ or $h_\mu^-$. We also prove that both $h_\mu^+$ and $h_\mu^-$ have an ergodic decomposition and we use it to prove a local Abramov formula for $h_\mu^-$.

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit circle, in various parameters, and applications to the study of quantum dynamics.

We obtain a family of metrics on the two-dimensional sphere whose geodesic flow is ergodic and Bernoulli. This family includes real analytic metrics.

In this paper we construct an example of a homeomorphism of the circle onto itself which is C∞, has no periodic points and no dense orbits. Moreover, the homeomorphism will have no more than two points of zero derivative. We alter this example to form a C∞ map of an interval to itself which has homtervals.