We show that Anosov systems in manifolds with trivial tangent bundles and with the property that the derivatives of the return maps at periodic orbits are multiples of the identity in the stable and unstable bundles are locally rigid. That is, any other smooth map, in a C^1 neighborhood such that it has the same Jordan normal form at corresponding periodic orbits is smoothly conjugate to it. This generalizes results of Castro and Moriyón (1997). We present several arguments for the main results. In particular, we use quasi-conformal regularity theory. We also extend the examples of an earlier paper of the author (1992) to show that some of the hypotheses we make in this paper are indeed necessary.

Let f:M\rightarrow M be a C^2 diffeomorphism of a compact Riemannian manifold of dimension m\geq 2 leaving invariant an ergodic Sinai–Ruelle–Bowen measure \mu with non-zero Lyapunov exponents. We prove that \mu can be approximated by ergodic measures supported on hyperbolic horseshoes with arbitrarily large unstable dimensions.

Given a twist map on the torus, we prove that the non-existence of a rotational invariant circle implies that the shear-rotation interval is non-trivial, i.e. the existence of orbits with non-zero mean acceleration. Under the same assumption, we prove the existence of infinitely many periodic orbits and positive entropy invariant measures with non-zero shear-rotation number. The method is variational, but relies on topological understanding of the shear-rotation interval of torus homeomorphisms homotopic to a Dehn twist.

Minimal homeomorphisms on the locally compact Cantor set are investigated. We prove that scaled dimension groups modulo infinitesimal subgroups determine topological orbit equivalence classes of locally compact Cantor minimal systems. We also introduce several full groups and show that they are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy. These are locally compact versions of the famous results for Cantor minimal systems obtained by Giordano et al. Moreover, proper homomorphisms and skew product extensions of locally compact Cantor minimal systems are examined and it is shown that every finite group can be embedded into the group of centralizers trivially acting on the dimension group.

We associate to each non-singular action A of a connected Lie group G on a closed manifold M and each A-invariant probability measure \mu on M, a linear mapping P_A from the cohomology H^*(\mathcal{F}) of the foliation \mathcal{F} given by the orbits of A onto the cohomology H^*(\mathcal{G}) of the Lie algebra \mathcal{G} of G. This extends to actions the asymptotic cycles of S. Schwartzman and D. Sullivan and gives a strong connection between the ergodic properties of A and the characteristic mapping \chi of \mathcal{F} as defined by dos Santos (Contemp. Math.161 (1994), 41–57). The ergodic notion of E-action introduced by Arraut and dos Santos (Topology31 (1992), 545–555) extends naturally to actions of Lie groups and we prove the main result of this article which shows that a non-singular E-action A of a nilpotent Lie group G completely determines the characteristic mapping \chi of the foliation \mathcal{F} given by A. This shows the strength of the above connection and gives as a corollary the vanishing theorem for foliated bundles proved by dos Santos.

We prove a generalized version of the Lefschetz fixed point theorem, and use it to obtain a variety of periodic and aperiodic solutions for differential delay equations; in particular, of the type \dot x(t) = f(x(t-1)). Here f:\mathbb R \to \mathbb R is odd and two-periodic, and we obtain both strictly periodic solutions and solutions periodic modulo a multiple of two. The qualitative behavior of solutions can be coded by symbol sequences containing the sequence of levels about which these solutions oscillate.

The linear action of SL (n, \mathbb{Z}) on \mathbb{R}^n has non-trivial invariant sets and matrices of SL (n, \mathbb{Z}) preserve volume. Here we prove that a Borel measure on \mathbb{R}^n, locally finite at a point, invariant under the linear action of SL (n, \mathbb{Z}) annihilating the set \{tz: t\geq 0,z\in \mathbb{Z}^n\} must be a scalar multiple of the Lebesgue measure. Our result extends a theorem due to S. G. Dani concerning locally finite measures, ergodic under the linear action of SL (n, \mathbb{Z}). Our approach is alternative, it exploits the Euclidean structure of the space and the fact that the probability of n integers being relatively prime is strictly positive.

We construct a Teichmüller space for the C^{1+}-conjugacy classes of hyperbolic dynamical systems on surfaces. After introducing the notion of an HR structure which associates an affine structure with each of the stable and unstable laminations, we show that there is a one-to-one correspondence between these HR structures and the C^{1+}-conjugacy classes. As part of the proof we construct a canonical representative dynamical system for each HR structure. This has the smoothest holonomies of any representative of the corresponding C^{1+}-conjugacy class. Finally, we introduce solenoid functions and show that they provide a good Teichmüller space.

We present some results of KAM type, comparable to the KAM theory for volume-preserving maps and flows, for generalized Hamiltonian systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd dimensional. Applications to the perturbation of three-dimensional steady Euler fluid particle path flows are considered with respect to the existence problem of barriers to fluid transport and mixing.

In this paper we consider symplectic and volume-preserving diffeomorphisms of compact manifolds, and prove that, C^{r} generically, 1\leq r\leq\infty, the invariant manifolds of hyperbolic periodic points contain a dense subset of recurrent points. As a consequence, each of these invariant manifolds is contained in its omega limit set, and its closure is a chain transitive set.

Let M be a three-dimensional compact connected oriented manifold and f : M \rightarrow M an expansive diffeomorphism. We prove that non-wandering points have local stable or unstable sets locally separating M. This property allows us to prove that if \Omega(f)=M then f is conjugate to a linear Anosov diffeomorphism and M=T^3, the three-dimensional torus.

For any diffeomorphism f of the two torus T^2 isotopic to the identity, the Ruelle invariant {\mathcal R}_\mu(f) is shown to vanish for some f-invariant probability measure \mu.

We establish a version of the local product structure (weak local product structure) for ergodic measures \overline{\mu} which are the invertible extension of ergodic weak Gibbs measures \mu invariant under piecewise C^0-invertible (infinite to one) Markov maps T. As a special case, \overline{\mu} possesses asymptotically ‘almost’ local product structure in the sense of Barreira, Pesin and Schmeling. Under piecewise conformality of T and the existence of a piecewise smooth representation of the dual map of T, the weak local product structure allows one to show that the pointwise dimension of \overline{\mu} exists almost everywhere and is the sum of the pointwise dimension of \mu and the pointwise dimension of the dual of \mu. Our results can be applicable to a natural extension of piecewise conformal two-dimensional Markov map which is related to a complex continued fraction and admits indifferent periodic points.

We use methods from conformal geometry to give an explicit estimate for how many iterations of Newton's method it takes, at most, to find all the roots of an arbitrary complex polynomial of fixed degree with prescribed precision.

We exhibit an example of a continuous map of the interval of type 2^\infty, and hence of zero topological entropy, for which the set of periodic points is not a G_\delta set. This disproves a conjecture by Sharkovsky from 1965. Unfortunately, this conjecture has been incorrectly quoted as a true statement by other authors in many papers and books.

Let \mathbb{M} be a monoid (e.g. \mathbb{N}, \mathbb{Z}, or \mathbb{M}^D), and \mathcal{A} an abelian group. \mathcal{A}^\mathbb{M} is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism \mathfrak{F}:\mathcal{A}^\mathbb{M}\longrightarrow\mathcal{A}^\mathbb{M} that commutes with all shift maps. Let \mu be a (possibly non-stationary) probability measure on \mathcal{A}^\mathbb{M}; we develop sufficient conditions on \mu and \mathfrak{F} so that the sequence \{\mathfrak{F}^N\mu\}_{N=1}^\infty weak* converges to the Haar measure on \mathcal{A}^\mathbb{M} in density (and thus, in Cesàro average as well). As an application, we show that, if \mathcal{A}=\mathbb{Z}_{/p} (p prime), \mathfrak{F} is any ‘non-trivial’ LCA on \mathcal{A}^{(\mathbb{Z}^D)}, and \mu belongs to a broad class of measures (including most Bernoulli measures (for D \geq 1) and ‘fully supported’ N-step Markov measures (when D=1)), then \mathfrak{F}^N\mu weak* converges to the Haar measure in density.

For a class of one-dimensional maps with two indifferent fixed points of the same order, a parallel to the arc-sine law for the number of positive partial sums of a real random walk is proved.

Let g be a polynomial automorphism of \mathbb{C}^2. We study the Hausdorff dimension and topological dimension of the Julia set of g. We show that when g is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is strictly greater than its topological dimension. Moreover, the Julia set cannot be locally connected. We also provide estimates for the dimension of the Julia sets in the general (not necessarily hyperbolic) case.

In a note which we received from W. Schachermayer he called attention to some confusion in his paper [S1], which he said was noticed by H. Buhler. Our reading of this note made us unhappily aware that some related confusion occurred in our papers [Sm] and [FS]. The purpose of the present note is to eliminate the latter confusion. We are grateful to Schachermayer for bringing the confusion to our attention. His original note, expanded, will be published as [S2].