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.

We study frequently hypercyclic operators, a natural new concept in hypercyclicity that was recently introduced by F. Bayart and S. Grivaux. We derive a strengthened version of their Frequent Hypercyclicity Criterion, which allows us to obtain examples of frequently hypercyclic operators in a straightforward way. Moreover, Bayart and Grivaux have noted that the frequent hypercyclicity setting differs from general hypercyclicity in that the set of frequently hypercyclic vectors need not be residual. We show here that, under weak assumptions, this set is only of first category. Motivated by this we study the question of whether one may write every vector in the underlying space as the sum of two frequently hypercyclic vectors. This investigation leads us to the introduction of a new notion, that of Runge transitivity.

We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤk and We show that they are either Haar measures or that every element of the action has zero metric entropy.

Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we define

where B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define

.

Two Cr, r ≥ 2, expanding maps of the circle which are absolutely continuously conjugate are Cr conjugate. Here ƒ and g: S1 → S1 are expanding if they stretch tangent vectors in some metric, and a conjugacy is an isomorphism h: S1 → S1 such that fh = hg.

The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as coupled phase oscillators. In this paper, a bifurcation structure of the infinite-dimensional Kuramoto model is investigated. A purpose here is to prove the bifurcation diagram of the model conjectured by Kuramoto in 1984; if the coupling strength $K$ between oscillators, which is a parameter of the system, is smaller than some threshold ${K}_{c} $, the de-synchronous state (trivial steady state) is asymptotically stable, while if $K$ exceeds ${K}_{c} $, a non-trivial stable solution, which corresponds to the synchronization, bifurcates from the de-synchronous state. One of the difficulties in proving the conjecture is that a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, has the continuous spectrum on the imaginary axis. Hence, the standard spectral theory is not applicable to prove a bifurcation as well as the asymptotic stability of the steady state. In this paper, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid the continuous spectrum on the imaginary axis. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition consisting of a countable number of eigenfunctions on a space of generalized functions. The semigroup generated by the linear operator will be estimated with the aid of the spectral theory on a rigged Hilbert space to prove the linear stability of the steady state of the system. The center manifold theory is also developed on a space of generalized functions. It is proved that there exists a finite-dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimension because of the continuous spectrum on the imaginary axis. These results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto.

Using the asymptotic properties of products of random matrices we study some properties of the subgroups of the linear group. These properties are centered around the theorem of J. Tits giving the existence of free subgroups in linear groups.

Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following:

(1) There is a set CT such that the ω-limit of Lebesgue-a.e. point equals CT. CT is a finite union of closed intervals or it coincides with the closure of the critical orbit.

(2) There is a constant λT such that for Lebesgue-a.e. x.

(3) λT > 0 if and only if T has an absolutely continuous invariant measure of positive entropy.

(4) , i.e. uniform hyperbolicity on periodic points implies λT > 0, and λT < 0 implies the existence of a stable periodic orbit.

Using elementary methods we show that for every unitary representation π7 of G = SL(2, ℝ) with no non-zero invariant vectors the matrix coefficients 〈 π(a(t))v, w 〉 of

decay exponentially fast for any vectors v, w Holder continuous in the direction of the rotation subgroup of G.

It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on ${\ell }^{p} ( \mathbb{Z} )$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is $ \mathcal{U} $-frequently hypercyclic but not frequently hypercyclic, and that there exists an operator which is frequently hypercyclic but not distributionally chaotic. These (surprising) counterexamples are given by weighted shifts on ${c}_{0} $. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.

A highly developed branch of the modern theory of dynamical systems is the study of deterministic ones with statistical properties in behaviour. During the last decade such systems were discovered in various domains of physics, chemistry, biology and technology. It is due to their complexity, that only the simplest of such systems have been analytically investigated (Lorenz system, Rikitake dynamo, billiard systems), and that is why numerical methods are widely used, especially in applied investigations. In numerical modelling we have no true trajectory of a dynamical system f, but an approximation = (x1, x2,…) such that the sequence of distances is small in some sense. For the case of round off errors in computer modelling, such a sequence is uniformly small, i.e. there exists some ε > 0, such that supnρ(xn+1, fxn)<ε. The sequence in this case is called an ε-trajectory of the dynamical system f[1]. In a series of investigations [1–14] a study was made of the properties and applications of ε-trajectories.

Let X be a compact metric space, f a continuous transformation on X, and Y a vector space with linear compatible metric. Denote by M(X) the collection of all the probability measures on X. For a positive integer n, define the nth empirical measure $L_{n}:X\mapsto M(X)$ as

\[ L_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}\delta_{f^{k}x}, \]

where $\delta_{x}$ denotes the Dirac measure at x. Suppose $\Xi:M(X)\mapsto Y$ is continuous and affine with respect to the weak topology on M(X). We think of the composite

\[ \Xi\circ L_{n}:X \xrightarrow{L_n} M(X)\xrightarrow{\Xi} Y \]

as a continuous and affine deformation of the empirical measure Ln. The set of divergence points of such a deformation is defined as

\[ D(f,\Xi)=\{x\in X\mid \mbox{the limit of }\Xi L_n x \mbox{ does not exist}\}. \]

In this paper we show that for a continuous transformation satisfying the specification property, if $\Xi(M(X))$ is a singleton, then set of divergence points is empty, i.e. <formula form="inline" disc="math" id="ffm008"><formtex notation="AMSTeX">$D(f,\Xi)=\emptyset$, and if $\Xi(M(X))$ is not a singleton, then the set of divergence points has full topological entropy, i.e.

\[ h_{\rm top}(D(f,\Xi))=h_{\rm top}(f). \]

We prove that those non-uniformly hyperbolic maps and flows (with singularities) that enjoy the K-property are also Bernoulli. In particular, many billiard systems, including those systems of hard balls and stadia that have the K-property, and hyperbolic billiards, such as the Lorentz gas in any dimension, are Bernoulli. We obtain the Bernoulli property for both the billiard flows and the associated maps on the boundary of the phase space.

Numerical-valued cohomological index theories for G-pairs (X, A) over B, where G is a compact Lie group, have proved useful in critical point theory and in proving Borsuk—Ulam and Bourgin—Yang theorems. More information (which is lost in taking numerical values) is obtained using an ideal-valued theory, and this theory is applied to estimating the size of the zero set of a G-map from certain G-manifolds to a G-module. Parametrized versions of these theorems are also obtained by a principle which applies quite generally.

In this survey we offer an overview of the so-called local entropy theory, which has been in development since the early 1990s. While doing so, we emphasize the connections between the topological dynamics and the ergodic theory points of view.

Let φ: ℝ × M → M be a continuous flow on a compact C∞ two-manifold M. It is proved that there exists a C1 flow ψ on M which is topologically equivalent to φ, and that the following conditions are equivalent:

(a) any minimal set of φ is trivial;

(b) φ is topologically equivalent to a C2 flow;

(c) φ is topologically equivalent to a C∞ flow.

Also proved is a structure and an existence theorem for continuous flows with non-trivial recurrence.

We study the dynamical properties of ergodic toral autmorphisms that have some eigenvalues of modulus one. For such automorphisms, all sufficiently fine smooth partitions generate measurably, but never topologically, and are never weak Bernoulli. The points of period k become uniformly distributed exponentially fast, and Lipschitz functions mix exponentially fast. Every reasonably smooth compact null set has the property that there is a dense set of periodic points whose entire orbit misses the set, but this is false for general compact null sets. Katznelson's property of almost weak Bernoulli can be strengthened to a certain exponential rate of independence, but breaks down at a critical number. Finally, open sets have return times that decay exponentially fast.

We describe the construction of almost-periodic solutions for a particular Schrödinger equation on a finite x- interval, depending on some potential V. The construction proceeds by iterating a KAM theorem about the existence of quasi-periodic solutions for some partial differential equations. As a result, we find for ‘almost all’ potentials V uncountably many almost-periodic solutions accumulating at the zero solution.

A theorem is proved giving a condition under which certain standing wave solutions of non-linear Schrödinger-type equations are linearly unstable. The eigenvalue equations for the linearized operator at the standing wave can be analysed by dynamical systems methods. A positive eigenvalue is then shown to exist by means of a shooting argument in the space of Lagrangian planes. The theorem is applied to a situation arising in optical waveguides.