In this paper we consider homeomorphisms $f:T^2 \rightarrow T^2$ homotopic to the identity and their rotation sets $\rho (\tilde{f})$, which are compact convex subsets of the plane. We show that if $\rho (\tilde{f})$ has an extremal point $(t,\omega )$ which is not a rational vector, then arbitrarily C0 close to f we can find a homeomorphism g such that $\rho (\tilde{g})\cap \rho (\tilde{f})^c\neq \emptyset$. So in this case, we have instability for the rotation set.

We consider dynamical systems on compact manifolds that are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that a (strong) integrability of the first hyperbolic time map implies a positive frequency of hyperbolic times. We also present an example of a map with a positive frequency of hyperbolic times at Lebesgue almost every point, but whose first hyperbolic time map is not integrable with respect to the Lebesgue measure.

Let $(X,\mathcal{B},\mu, T)$ be an ergodic dynamical system on the finite measure space $(X,\mathcal{B},\mu,T)$ and $\mathcal{K}$ its Kronecker factor. We denote by U the restriction of T onto $\mathcal{K}^{\perp}$ the orthocomplement of $\mathcal{K}$. We give a spectral characterization in L2 of Wiener–Wintner functions in terms of the capacity of the support of the maximal spectral type of U and the almost everywhere continuity of the fractional rotated ergodic Hilbert transform. The study of the L2 case leads to new classes of dynamical systems.

This paper explores the effect of various graphical constructions upon the associated graph C*-algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that out-splittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C*-algebras. We generalize the notion of a delay as defined in (D. Drinen, Preprint, Dartmouth College, 2001) to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C*-algebras. We provide examples which suggest that our results are the most general possible in the setting of the C*-algebras of arbitrary directed graphs.

We consider piecewise monotone maps of the interval with zero entropy or no periodic points. First, we give a rigid model for these maps: the interval translations mappings, possibly with flips. It follows, for example, that the complexity of a piecewise monotone map of the interval is at most polynomial if and only if this map has a finite number of periodic points up to monotone equivalence. Second, we study the invariant and ergodic measures of a piecewise monotone map with zero entropy and prove that their number is bounded by twice the number of monotony intervals; for a piecewise increasing map their number is at most the number of intervals.

We prove that a rank-one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences, in the sense that the mean ergodic theorem holds for a family of what we call dynamical sequences. The application of our theorem shows that the class of polynomial rank-one transformations, rank-one transformations where the spacers are chosen to be the values of a polynomial with some mild conditions on the polynomials, that have restricted growth are mixing transformations, implying, in particular, Adams' result on staircase transformations. Another application yields a new proof that Ornstein's class of rank-one transformations constructed using ‘random spacers’ are almost surely mixing transformations.

We prove several new results about AF-equivalence relations and relate these to Cantor minimal systems (i.e. to minimal Z-actions). The results we obtain turn out to be crucial for the study of the topological orbit structure of more general countable group actions (as homeomorphisms) on Cantor sets, which will be the topic of a forthcoming paper. In all this, Bratteli diagrams and their dynamical interpretation are indispensable tools.

One of the corollaries of Ornstein's isomorphism theorem is that if $(Y,S,\nu)$ is an invertible measure-preserving transformation and $(Y,S^2,\nu)$ is isomorphic to a Bernoulli shift, then $(Y,S,\nu)$ is isomorphic to a Bernoulli shift. In this paper we show that non-invertible transformations do not share this property. We do this by exhibiting a uniformly two-to-one endomorphism $(X,\sigma,\mu)$ which is not isomorphic to the one-sided Bernoulli two shift. However, $(X,T^2,\mu)$ is isomorphic to the one-sided Bernoulli four shift.

We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0<\gamma < 1$. We prove by showing both lower and upper bounds that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions. We also show that the density function h of $\mu$ has the order $x^{-\gamma}$ as $x\to 0$. Perron–Frobenius operators are the main tool used for proofs.

We consider one-dimensional dynamical systems with indifferent fixed points (fixed points with derivative one). Many such maps have absolutely continuous ergodic infinite invariant measures. We study the limit of the ratio of the ergodic sum of fA to that of fB, where the integrals of fA and fB are infinite with respect to the absolutely continuous ergodic infinite invariant measure. If fA and fB are analytic functions on [0, 1], the result in this paper makes it clear whether the ratio of the ergodic sum of fA to that of fB converges in the Lebesgue measure or not.

Consider a wave equation (WE) with constant coefficients in $\mathbb{R}^n$ for even $n\ge 2$ and with variable coefficients for even $n\ge4$. We study the distribution $\mu_t$ of the random solution at time $t\in\mathbb{R}$. The initial probability measure $\mu_0$ has a translation-invariant covariance, zero mean and finite mean density for the energy. It also satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central Limit Theorem (CLT) for the WE. The proof for the case of constant coefficients is based on stationary phase asymptotics of the solution in the Fourier representation and Bernstein's ‘room–corridor’ argument. The case of variable coefficients is reduced to that of constant ones by a version of the scattering theory, based on Vainberg's results on local energy decay.

A singular-hyperbolic attractor for vector fields is a partially hyperbolic attractor with singularities (that are hyperbolic) and volume expanding central direction. The geometric Lorenz attractor is the most representative example of a singular-hyperbolic attractor. In this paper, we prove that if $\Lambda$ is a singular-hyperbolic attractor of a three-dimensional vector field X, then there is a neighborhood U of $\Lambda$ in M such that every attractor in U of a Cr vector field Cr close to X is singular, i.e. it contains a singularity. With this result we prove the following corollaries. There are neighborhoods U of $\Lambda$ (in M) and $\mathcal U$ of X (in the space of Cr vector fields) such that if n denotes the number of singularities of X in $\Lambda$, then $\#\{A\subset U:A$ is an attractor of $Y\in\mathcal U\}\leq n$. Every three-dimensional vector field Cr close to one exhibiting a singular-hyperbolic attractor has a singularity non-isolated in the non-wandering set. A singularity of a three-dimensional Cr vector field Y is stably non-isolated in the non-wandering set if it is the unique singularity of a singular-hyperbolic attractor of Y. These results generalize well-known properties of the geometric Lorenz attractor.

In the 1970s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system with a perturbation of size $\varepsilon$, the actions linked to the unperturbed Hamiltonian vary only by the order of $\varepsilon^b$ over a time of the order of $\exp (C\varepsilon^{-a})$ for some positive constants a, b and C, provided that the unperturbed Hamiltonian meets some generic transversality condition known as steepness. Among steep systems, convex or quasiconvex systems are easier to describe since the use of energy conservation allows the proof of exponential estimates of stability to be shortened. In this case, Lochak–Neishtadt and Poschel have independently obtained the stability exponents a = b = 1/2n for systems of n degrees of freedom—especially the time exponent (a) is expected to be optimal (see P. Lochak, J.-P. Marco and D. Sauzin. Preprint. Institut de Máthematique de Jussieu, 1999; J.-P. Marco and D. Sauzin. Preprint. Publ. Math. Inst. Hautes Etudes Science, 2001). Moreover, Lochak's study relies on simultaneous Diophantine approximation which gives a very transparent proof.

However, the proof in the steep case has rarely been studied since Nekhorochev's original work despite various physical examples where the model Hamiltonian is only steep. Here, we combine the original scheme with a simultaneous Diophantine approximation as in Lochak's proof. This yields significant simplifications with respect to Nekhorochev's reasoning: it also allows the exponents $a=b=(2n p_1\dotsb p_{n-1})^{-1}$ where $(p_1\dotsb p_{n-1})$ are the steepness indices of the considered Hamiltonian to be obtained. In the quasiconvex case, the steepness indices are all equal to one and we find the same exponents 1/2n as Lochak–Neishtadt and Poschel, whose results are thus generalized to the steep case.

It is well known that the action of a hyperbolic element (‘cat map’) of the modular group on the 2-torus has strong chaotic dynamical properties such as mixing and exponential decay of correlations. In this note we study stability of this behaviour with respect to kicks. Our approach is based on geometric group theory, and in particular on a new result on quasi-morphisms of the modular group.

We investigate normalizers of full groups of a topological dynamical system by means of cocycles. Some of the investigations are conducted in connection with index maps, the centralizer of a topological dynamical system and groups of automorphisms of a C*-crossed product.