In this paper we obtain a polynomial error term for the closed orbit counting function associated to certain hyperbolic flows. In the case of weak-mixing transitive Anosov flows no further conditions are required; for general weak-mixing hyperbolic flows a diophantine condition on the periods of the closed orbits is required.

We prove L^2 variation inequalities for operators defined by the convolution powers of probability measures on locally compact Abelian groups. In some cases we also obtain L^p results for 1<p<\infty. These inequalities imply the pointwise convergence of these operators and give an estimate of the number of upcrossings.

The dynamics of a shift-like mapping f(x,y,z)=(y,z,yz+by+cz+dx+e) (d≠0) is studied, which is one of the normal forms of irregular quadratic polynomial automorphisms of [Copf ]3. It is shown that f has Green functions G±(x,y,z) associated to the dynamical degree of f. To obtain them, we consider a sort of filtration property.

We consider the action of the (n-1)-dimensional group of diagonal matrices in SL(n,\mathbb{R}) on SL(n,\mathbb{R})/\Gamma, where \Gamma is a lattice and n\ge 3. Far-reaching conjectures of Furstenberg, Katok–Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for \Gamma={\rm SL}(n,\mathbb{Z}) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n=2, such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.

We prove certain rigidity properties of higher-rank abelian product actions of the type \alpha\times \mathop{\rm Id}\nolimits_N:{\mathbb Z}^\kappa\to\operatorname{Diff}(M\times N), where \alpha is (TNS) (i.e. is hyperbolic and has some special structure of its stable distributions). Together with a result about product actions of property (T) groups, this implies the local rigidity of higher-rank lattice actions of the form \alpha\times \mathop{\rm Id}\nolimits_{\mathbb{T}}:\Gamma\to\operatorname{Diff}(M\times\mathbb{T}), provided \alpha has some rigidity properties itself, and contains a (TNS) subaction.

Given a two-parameter family of three-dimensional diffeomorphisms \{f_{a,b}\}_{a,b}, with a dissipative (but not sectionally dissipative) saddle fixed point, assume that a special type of quadratic homoclinic tangency of the invariant manifolds exists. Then there is a return map f^n_{a,b}, near the homoclinic orbit, for values of the parameter near such a tangency and for n large enough, such that, after a change of variables and reparametrization depending on n, this return map tends to a simple quadratic map. This implies the existence of strange attractors and infinitely many sinks as in other known cases. Moreover, there appear attracting invariant circles, implying the existence of quasi-periodic behaviour near the homoclinic tangency.

We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular, we discuss the role such a choice plays with regard to matching rules for such structures.

In this paper we show that strong generalizations of the measurable Livšic theorem for cocycles taking values in connected non-compact linear semisimple Lie groups, a canonical example being SL(2, \mathbb{R}), can be deduced from an elegant approach of Brin and Pesin to the dynamics of partially hyperbolic systems.

It will be proved that some topological conjugacy invariants, called K-groups and Bowen–Franks groups, for subshifts are invariants for flow equivalence by an argument of symbolic dynamical systems. Some examples of non-sofic subshifts that are not flow equivalent will be presented.

We prove the following property: if X in \mathcal G^1(M) has no singularity and x \in \Sigma(X), then \overline{\operatorname{orbit}(x)} \cap \overline{\operatorname{per}(X)} \not = \emptyset. In addition, if we assume \overline{\operatorname{per}_i(X)} \cap \overline{\operatorname{per}_j(X)} = \emptyset for i \not = j, then \overline{\operatorname{per}(X)} = \bigcup_{i=0}^{n-1} \overline{\operatorname{per}_i(X)} is a hyperbolic set. Moreover, we shall give a proof of the \Omega-stability conjecture for flows.

Lorenz maps are dynamical systems on the interval related to the Lorenz flow. Monotone Lorenz families will be introduced and it will be shown that all possible topological dynamical behavior of Lorenz maps is realized within every monotone Lorenz family. Moreover, the structure of the parameter space for such families is described.

We consider dynamics of sub-hyperbolic and semi-hyperbolic semigroups of rational functions on the Riemann sphere and will show some no wandering domain theorems. The Julia set of a rational semigroup in general may have non-empty interior points. We give a sufficient condition that the Julia set has no interior points. From some information about forward and backward dynamics of the semigroup, we consider when the area of the Julia set is equal to zero or an upper estimate of the Hausdorff dimension of the Julia set.

In the planar N-body problem, N point masses move in the plane under their mutual gravitational attraction. It is classical that the dynamics of this motion conserves the integrals of motion: center of mass, linear momentum, angular momentum c, and energy h. Further, the motion has a rotational symmetry. The dynamics thus takes place on a (4N-7)-dimensional open manifold, known as the reduced integral manifold [mfr ]R(M,ν). The topology of this manifold depends only on the masses M = (m1,...,mN) and the quantity ν = -hc2. In spite of the central importance of this manifold in a classical dynamical problem, very little is known about the topology of [mfr ]R(M,ν). In this note, we build on the topological analysis of Smale to describe the homology of [mfr ]R(M,ν). A variety of homological results are presented, including the computation of the homology groups for ν very large for all M; and for all ν for three masses, and for four equal masses.

The aim of this paper is to present a one-parameter family X_a of three-dimensional vector fields which has infinitely many persistent Hénon-like strange attractors. This family also has infinitely many periodic attractors for values of the parameter in an open set.

Relative periodic solutions are ubiquitous in dynamical systems with continuous symmetry. Recently, Sandstede, Scheel and Wulff derived a center bundle theorem, reducing local bifurcation from relative periodic solutions to a finite-dimensional problem. Independently, Lamb and Melbourne showed how to systematically study local bifurcation from isolated periodic solutions with discrete spatiotemporal symmetries.

In this paper, we show how the center bundle theorem, when combined with certain group theoretic results, reduces bifurcation from relative periodic solutions to bifurcation from isolated periodic solutions. In this way, we obtain a systematic approach to the study of local bifurcation from relative periodic solutions.

Let X, Y be topologically mixing subshifts of finite type and \pi : X \rightarrow Y a factor map. For each \alpha \geq 0, the weighted entropy function \phi_{\alpha} is defined by \phi_{\alpha} (\mu) = h (\mu) + \alpha h (\pi \mu) for each invariant measure \mu on X. To investigate whether for a given \alpha > 0 there is a unique measure which achieves \sup_{\mu} \phi_{\alpha} (\mu), we use the concept of compensation functions which was first considered by Boyle and Tuncel and has been developed by Walters. We prove that if there is a certain kind (more general than summable variation) of compensation function, then for each \alpha \geq 0 the shift-invariant measure which maximizes the weighted entropy is unique. In particular, if the compensation function is locally constant, then the unique measure is Markov and mixing. We classify the 1-block codes from a 3-symbol subshift of finite type to a 2-symbol subshift in terms of what type of compensation function exists or does not exist, providing examples of factor maps which do and do not satisfy the hypothesis. Also we study general properties of compensation functions and the maximal weighted entropy map as a function of the weight.

We construct, for each irrational number \alpha, a minimal C^1-diffeomorphism of the circle with rotation number \alpha which is not ergodic with respect to the Lebesgue measure.

In our paper [1], in the proof of Proposition 1, we implicitly assume that the polynomial P is monic, although later we apply this proposition to polynomials which are not monic. The following corrections should be made in the proof of Proposition 1.

We consider measurable conjugacies between two non-solvable subgroups G_1, G_2 of the group of analytic diffeomorphisms of the circle which admit a finite generating set close to the identity. These measurable conjugacies are characterized by our main result which asserts that, up to finite coverings, they coincide almost everywhere with an analytic diffeomorphism of the circle.