Extending a result of Livsic [10] it is proved that the coboundary equation f(Tx)−f(x) = g(x) admits a C∞ solution f for C∞g when T is an ergodic toral endomorphism and g sums to zero over every periodic orbit. The same statement is false with C1 in place of C∞, in contrast to the Livsic (hyperbolic) theorem. In one dimension the ‘Lip α’ case leads to questions relating to the generalized Riemann hypothesis.

We prove that for any self-conformal measures, without any separation conditions, the multifractal formalism partially holds. The result follows by establishing certain Gibbs properties for self-conformal measures.

We consider a class of Gibbs measures on self-affine Sierpiński carpets and perform the multifractal analysis of its elements. These deterministic measures are Gibbs measures associated with bundle random dynamical systems defined on probability spaces whose geometrical structure plays a central role. A special subclass of these measures is the class of multinomial measures on Sierpiński carpets. Our result improves the already known result concerning the multifractal nature of the elements of this subclass by considerably weakening and in some cases even eliminating a strong separation condition of geometrical nature.

We prove that every convex polygon in ℝ2 with vertices in ℚ2 is a rotation set for some isotopic to identity homeomorphism of the two dimensional terms.

According to a conjecture of Lindenstrauss and Tsukamoto, a topological dynamical system $(X,T)$ is embeddable in the $d$-cubical shift $(([0,1]^{d})^{\mathbb{Z}},\text{shift})$ if both its mean dimension and periodic dimension are strictly bounded by $d/2$. We verify the conjecture for the class of systems admitting a finite-dimensional non-wandering set and a closed set of periodic points. This class of systems is closely related to systems arising in physics. In particular, we prove an embedding theorem for systems associated with the two-dimensional Navier–Stokes equations of fluid mechanics. The main tool in the proof of the embedding result is the new concept of local markers. Continuing the investigation of (global) markers initiated in previous work it is shown that the marker property is equivalent to a topological version of the Rokhlin lemma. Moreover, new classes of systems are found to have the marker property, in particular, extensions of aperiodic systems with a countable number of minimal subsystems. Extending work of Lindenstrauss we show that, for systems with the marker property, vanishing mean dimension is equivalent to the small boundary property.

We define, following Veech, the Fuchsian group $\Gamma(P)$ of a rational polygon $P$. If $P$ is simply-connected, then ‘rational’ is equivalent to the condition that all interior angles of $P$ be rational multiples of $\pi$. Should it happen that $\Gamma(P)$ has finite covolume in $\mathop{\rm PSL}\nolimits (2, {\Bbb R})$ (and is thus a {\it lattice}), then a theorem of Veech states that every billiard path in $P$ is either finite or uniformly distributed in $P$.

We consider the Fuchsian groups of various rational triangles. First, we calculate explicitly the Fuchsian groups of a new sequence of triangles, and discover they are lattices. Interestingly, the lattices found are not commensurable with those previously known. We then demonstrate a class of triangles whose Fuchsian groups are {\it not\/} lattices. These are the first examples of such triangles. Finally, we end by showing how one may specify algebraically, i.e. by an explicit polynomial in two variables, the Riemann surfaces and holomorphic one-forms that are associated to a simply-connected rational polygon. Previously, these surfaces were known by their geometric description. As an example, we show a connection between the billiard in a regular polygon and the well-known Fermat curves of the algebraic equation $x^n + y^n = 1$.

We show that every shift of finite type $X$ with positive entropy has proper subshifts of finite type with entropy strictly smaller than the entropy of $X$, but with entropy arbitrarily close to the entropy of $X$. Consequently, $X$ contains an infinite chain of subshifts of finite type which is strictly decreasing in entropy.

We investigate the ergodic theory of Poisson suspensions. In the process, we establish close connections between finite and infinite measure-preserving ergodic theory. Poisson suspensions thus provide a new approach to infinite-measure ergodic theory. Fields investigated here are mixing properties, spectral theory, joinings. We also compare Poisson suspensions to the apparently similar looking Gaussian dynamical systems.

In this paper, we prove the following results about the shift map on the inverse limit space of a compact metric space and a sole bonding map: (1) the chain recurrent set of the shift map equals the inverse limit space of the chain recurrent set of the sole bonding map. Similar results are proved for the nonwandering set, ω-limit set, recurrent set, and almost periodic set. (2) The shift map on the inverse limit space is chaotic in the sense of Devaney if and only if its sole bonding map is chaotic. (3) If the sole bonding map is ω-chaotic, then the shift map on the inverse limit space is ω-chaotic. With a modification of the definition of ω-chaos we show the converse is true. (4) We prove that a transitive map on a closed invariant set containing an interval must have sensitive dependence on initial conditions on the whole set. Then it is both ω-chaotic and chaotic in the sense of Devaney with chaotic set the whole set. At last we give a new method of constructing chaotic homeomorphisms on chainable continua, especially on pseudo-arcs.

We consider a class of ordinary differential equations describing one-dimensional systems with a quasi-periodic forcing term and in the presence of large damping. We discuss the conditions to be assumed on the mechanical force and the forcing term for the existence of quasi-periodic solutions which have the same frequency vector as the forcing.

The existence of transversal ejection—collision orbits in the restricted three-body problem is shown to imply, via the KAM theorem, the existence, for certain intervals of (large) values of the Jacobi constant, of an uncountable number of invariant punctured tori in the corresponding (non-compact) energy surface. The proof is based on a comparison between Levi-Civita and McGehee regularizing variables. That these transversal ejection-collision orbits do actually exist was proved in [5] in the case where one of the primaries has a small mass and the zero-mass body revolves around the other (and for all values of the Jacobi constant compatible with the existence of three connected components for the Hill region); it is proved here without any restriction on the masses, well in the spirit of Conley's thesis [3].

Let h be the Hausdorff dimension of the Julia set of a rational map with no nonperiodic recurrent critical points. We give necessary and sufficient conditions for h-dimensional Hausdorff measure and h-dimensional packing measure of the Julia set to be positive and finite. We also show that either the Julia set is the whole Riemann sphere or h < 2 and that if a rational map (not necessarily with no nonperiodic recurrent critical points!) has a rationally indifferent periodic point, then h > 1/2.

Let g be the geodesic flow on the unit tangent bundle of a C3 compact surface of negative curvature. Let μ be the g-invariant measure of maximal entropy. Let h be a uniformly parametrized flow along the horocycle foliation, i.e., such a flow exists, leaves μ invariant, and is unique up to constant scaling of the parameter (Margulis). We show that any measure-theoretic conjugacy: (h, μ) → (h′, μ′) is a.e. of the form θ, where θ is a homeomorphic conjugacy: g → g′. Furthermore, any homeomorphic conjugacy g → g′; must be a C1 diffeomorphism.

This paper is based on the author's previous work (Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products. Ergod. Th. & Dynam. Sys. 21 (2001), 563–603). We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup G generated by a compact family of rational maps on the Riemann sphere $\overline{\mathbb C}$ of degree two or greater, the Julia set of any subsemigroup of G is uniformly perfect. We define the semi-hyperbolicity of dynamics on fiber bundles and show that, if the dynamics on a fiber bundle is semi-hyperbolic, then the fiberwise Julia sets are porous, and the dynamics is weakly rigid. Moreover, we show that if the dynamics is semi-hyperbolic and the fiberwise maps are polynomials, then under some conditions, the fiberwise basins of infinity are John domains. We also show that the Julia set of a rational semigroup (a semigroup generated by rational maps on $\overline{\mathbb C}$) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.

For the family of exponential maps $E_{\kappa}(z)=\exp(z)+\kappa$, we prove an analog of Böttcher's theorem by showing that any two exponential maps $E_{\kappa_1}$ and $E_{\kappa_2}$ are conjugate on suitable subsets of their escaping sets, and this conjugacy is quasiconformal. Furthermore, we prove that any two attracting and parabolic exponential maps are conjugate on their sets of escaping points; in fact, we construct an analog of Douady's ‘pinched disk model’ for the Julia sets of these maps. On the other hand, we show that two exponential maps are generally not conjugate on their sets of escaping points. We also answer several questions about escaping endpoints of dynamic rays. In particular, we give a necessary and sufficient condition for the ray to be continuously differentiable in such a point, and show that escaping points can escape arbitrarily slowly. Furthermore, we show that the principle of topological renormalization is false for attracting exponential maps.

Measure equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field has grown considerably over the past few years, developing tools to distinguish between different ME classes of countable groups. On the other hand, constructions of ME groups are very rare. We present a new method, based on a notion of measurable free factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about ME for countable groups.

The celebrated Livšic theorem [A. N. Livšic, Certain properties of the homology of Y-systems, Mat. Zametki 10 (1971), 555–564; A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320] states that given a manifold M, a Lie group G, a transitive Anosov diffeomorphism f on M and a Hölder function η:M↦G whose range is sufficiently close to the identity, it is sufficient for the existence of ϕ:M↦G satisfying η(x)=ϕ(f(x))ϕ(x)−1 that a condition—obviously necessary—on the cocycle generated by η restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.

We obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of three-dimensional manifolds having compact center leaves: either there is a unique entropy-maximizing measure, this measure has the Bernoulli property and its center Lyapunov exponent is 0, or there are a finite number of entropy-maximizing measures, all of them with non-zero center Lyapunov exponents (at least one with a negative exponent and one with a positive exponent), that are finite extensions of a Bernoulli system. In the first case of the dichotomy, we obtain that the system is topologically conjugated to a rotation extension of a hyperbolic system. This implies that the second case of the dichotomy holds for an open and dense set of diffeomorphisms in the hypothesis of our result. As a consequence, we obtain an open set of topologically mixing diffeomorphisms having more than one entropy-maximizing measure.

A simpler treatment is given of the theorems of Dye and Krieger concerning the classification of non-singular ergodic transformations up to orbit equivalence.

We use the methods developed with Lyubich for proving complex bounds for real quadratics to extend de Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows.

As another application of our methods we present a new proof of a theorem of Petersen on local connectivity of some Siegel Julia sets.