The paper is devoted to the triangular maps of the square into itself. The results presented were recently obtained by the author and are briefly stated (in Russian) in a difficult paper as well as those (jointly published with A. N. Sharkovsky) published in ECIT-89 (abstract). All these results are systematized and extended by the new ones. The more detailed proofs of all statements are given. It is shown, for example, that triangular maps exist such that their minimal attraction centres do not coincide with the centres, as well as such ones exist that the Milnor attractor is not contained in the closure of the set of periodic points.

‘Bunching’ conditions on an Anosov system guarantee the regularity of the Anosov splitting up to C2−ε. Open dense sets of symplectic Anosov systems and geodesic flows do not have Anosov splitting exceeding the asserted regularity. This is the first local construction of low-regularity examples.

Let G be a Lie group and Γ be a discrete subgroup. We show that if {μn} is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G.

This paper discusses the relations between the following properties o finite measure preserving ergodic actions of a countable group G: strong ergodicity (i.e. the non-existence of almost invariant sets), uniqueness of G-invariant means on the measure space carrying the group action, and certain cohomological properties. Using these properties one can characterize all actions of amenable groups and of groups with Kazhdan's property T. For groups which fall in between these two definations these notions lead to some interesting examples.

One may often decompose the domain of a topologically transitive map into finitely many regular closed pieces with nowhere dense overlap in such a way that these pieces map into one another in a periodic fashion. We call decompositions of this kind regular periodic decompositions and refer to the number of pieces as the length of the decomposition. If $f$ is topologically transitive but $f^{n}$ is not, then $f$ has a regular periodic decomposition of some length dividing $n$. Although a decomposition of a given length is unique, a map may have many decompositions of different lengths. The set of lengths of decompositions of a given map is an ideal in the lattice of natural numbers ordered by divisibility, which we call the decomposition ideal of $f$. Every ideal in this lattice arises as a decomposition ideal of some map. Decomposition ideals of Cartesian products of transitive maps are discussed and used to develop various examples. Results are obtained concerning the implications of local connectedness for decompositions. We conclude with a comprehensive analysis of the possible decomposition ideals for maps on 1-manifolds.

We present an analysis of one-dimensional models of dynamical systems that possess ‘coherent structures’: global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron–Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron–Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for non-invertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalize the notions of almost-invariant and almost-cyclic sets to non-autonomous dynamical systems and provide a new ensemble-based formalism for coherent structures in one-dimensional non-autonomous dynamics.

There is no algorithm which will take a description of a celluar automaton and determine whether it has zero topological entropy, or for any fixed ε > 0 compute its topological entropy to a tolerance e. Furthermore a set of aperiodic Wang tiles arising from Penrose's kite and dart tiles is used to demonstrate specific examples of cellular automata with a single periodic point but non-trivial non-wandering sets, which furthermore can be constructed to have arbitrarily high topological entropy.

We present an overview and some new applications of the approximation by conjugation method introduced by Anosov and Katok more than 30 years ago (Trans. Moscow Math. Soc.23 (1970), 1–35). Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi (Asterisque49 (1977), 37–59) and continuing with exotic invariant sets of rational maps of the Riemann sphere (J. London Math. Soc. (2) 34 (1986), 375–384) and the construction of invariant tori with non-standard and unexpected behavior in the context of KAM theory (Pitman Research Notes Mathematical Series 243 (1992); Proc. Int. Congr. Mathematicians (Berlin, 1998) Vol. 11, 797–808). Recently the method has been experiencing a revival. Some of the new results presented in the paper illustrate variety of uses for tools available for a long time, others exploit new methods, in particular the possibility of mixing in the context of Liouvillean dynamics discovered by the first author (Ergod. Th. & Dynam. Sys.22 (2002) 437–468; Proc. Amer. Math. Soc.130 (2002), 103–109).

It is shown how regular model sets can be characterized in terms of the regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map $\beta$ and then relate the properties of $\beta$ to those of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.

For any compact oriented surface $\Sigma$ we consider the group of diffeomorphisms of $\Sigma$ which preserve a given area form. In this paper we show that the vector space of homogeneous quasi-morphisms on this group has infinite dimension. This result is proved by constructing explicitly and for each surface an infinite family of independent homogeneous quasi-morphisms. These constructions use simple arguments related to linking properties of the orbits of the diffeomorphisms.

Let SA be an irreducible and aperiodic topological Markov chain. If SĀ is an irreducible and aperiodic topological Markov chain, whose topological entropy is less than that of SA, then there exists an irreducible and aperiodic topological Markov chain, whose topological entropy equals the topological entropy at SĀ, and that is a subsystem of SA. If S is an expansive homeomorphism of the Cantor discontinuum, whose topological entropy is less than that of SA, and such that for every j∈ℕ the number of periodic points of least period j of S is less than or equal to the number of periodic points of least period j of SA, then S is topological conjugate to a subsystem of SA.

In this paper a new proof of a theorem of Ruelle about real Perron-Frobenius type operators is given. This theorem is then extended to complex Perron-Frobenius type operators in analogy with Wielandt's theorem for matrices. Finally two questions raised by Ruelle and Bowen concerning analyticity properties of zeta functions for flows are answered.

We show that entire functions which are critically finite and which meet certain growth conditions admit ‘Cantor bouquets’ in their Julia sets. These are invariant subsets of the Julia set which are homeomorphic to the product of a Cantor set and the line [0, ∞). All of the curves in the bouquet tend to ∞ in the same direction, and the map behaves like the shift automorphism on the Cantor set. Hence the dynamics near ∞ for these types of maps may be analyzed completely. Among the entire maps to which our methods apply are exp (z), sin (z), and cos (z).

We introduce a class of vector fields on n-manifolds containing the hyperbolic systems, the singular-hyperbolic systems on 3-manifolds, the multidimensional Lorenz attractors and the robust transitive singular sets in Li et al [Robust transitive singular sets via approach of an extended linear Poincaré flow. Discrete Contin. Dyn. Syst.13(2) (2005), 239–269]. We prove that the closed orbits of a system in such a class are hyperbolic in a persistent way, a property which is false for higher-dimensional singular-hyperbolic systems. We also prove that the singularities in the robust transitive sets in Li et al are similar to those in the multidimensional Lorenz attractor. Our results will give a partial negative answer to Problem 9.26 in Bonatti et al [Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III). Springer, Berlin, 2005].

A measure-preserving transformation (respectively a topological system) is null if the metric (respectively topological) sequence entropy is zero for any sequence. Kushnirenko has shown that an ergodic measure-preserving transformation has a discrete spectrum if and only if it is null. We prove that for a minimal system this statement remains true modulo an almost one-to-one extension. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, some necessary conditions for a transitive non-minimal system to be null are obtained.

Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering a question in Blanchard et al (F. Blanchard, B. Host and A. Maass, Topological complexity. Ergod. Th. & Dynam. Sys., 20 (2000), 641–662). For a group action we give a direct proof of the fact that the factor induced by the smallest invariant equivalence relation containing the regionally proximal relation is equicontinuous. Furthermore, we show that a non-equicontinuous minimal distal system is not null.

Suppose q is a holomorphic quadratic differential on a compact Riemann surface of genus g ≥ 2. Then q defines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at most T is at most quadratic in T. An application is given to billiards.

This is an extended report on a problem session about geodesic flows held on May 30, 1984 at MSRI, Berkeley. The list of six authors reflects the principal participants in the discussion. Contributions of other participants are mentioned in the text. The session was chaired by A. Katok; this report was prepared by K. Burns in collaboration with A. Katok.

Let Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

The paper discusses the connectivity of periodic and preperiodic domains in the stable set in the iteration of a meromorphic function. The connectivity of an invariant component has one of the values 1, 2, ∞. Examples are constructed to show that the connectivity of a preperiodic component may take any value.

This paper discusses a division of one-dimensional cellular automata into classes which seem to correspond to different types of behaviour.