Using the techniques developed in Alves and Sousa Ramos (Comm. Math. Phys.204(1) (1999), 89–114), we generalize the Milnor and Thurston results about the zeta function, semiconjugacy and topological entropy of interval maps to tree maps.

In this paper, we give the explicit construction of certain components of the space of holomorphic foliations of codimension one, in complex projective spaces. These components are associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. Some of them, the so-called exceptional or Klein–Lie components, are rigid in the sense that all generic foliations in the component are equivalent (Example 1). In particular, we obtain rigid foliations of all degrees. Some generalizations and open problems are given at the end of §1.

We present new invariants for substitutional dynamical systems. Our main contribution is a flow invariant which is strictly finer than, but related and akin to, the dimension groups of Herman et al. We present this group as a stationary inductive limit of a system associated to an integer matrix defined from combinatorial data based on the class of special words of the dynamical system.

A smooth $\mathbb{Z}^p$-action on a closed orientable manifold M is cohomologically rigid (CR) if the cohomology of the induced $\mathbb{Z}^p$-module on the smooth functions on M is isomorphic to the real cohomology of the torus Tp. We prove some general properties of a smooth $\mathbb{Z}^p$-action whose first cohomology is isomorphic to Rp. When the manifold is a low-dimensional torus Tq, $1 \le q \le 2$, we prove that any minimal (all orbits are dense) smooth $\mathbb{Z}^p$-action on Tq whose first cohomology is isomorphic to Rp is $C^\infty$ conjugate to an affine $\mathbb{Z}^p$-action. As a corollary we show that the CR $\mathbb{Z}^p$-actions on Tq, $1 \le q \le 2$, are smooth conjugations of affine CR actions.

We introduce a non-commutative generalization of the notion of (approximately proper) equivalence relations and propose the construction of a ‘quotient space’. We then consider certain one-parameter groups of automorphisms of the resulting C*-algebra and prove the existence of KMS states at every temperature. In a model originating from thermodynamics we prove that these states are unique as well. We also show a relationship between maximizing measures (the analogue of the Aubry–Mather measures for expanding maps) and ground states. In the last section we explore an interesting example of phase transitions.

It is proved that special flows over irrational rotations and under functions whose Fourier coefficients are of order O(1/|n|) are disjoint in the sense of Furstenberg from all mixing flows. This is an essential strengthening of a classical result by Kočergin on the absence of mixing of special flows built over irrational rotations and under bounded variation roof functions.

For each of the following properties, there is an isometric generalized interval exchange transformation (i.e. isometric GIET) having such property: (a) non-trivial recurrence orbits are exceptional and the union of them is a dense set, moreover the intersection of the closure of two such orbits is the union of finite orbits; (b) coexistence of dense orbits and exceptional orbits; (c) existence of a dense sequence of exceptional orbits $\{\mathcal{O}(p_k)\colon k=1,2,\dotsc\}$ such that $\overline{\mathcal{O}(p_1)}\subsetneqq\overline{\mathcal{O}(p_2)}\subsetneqq\dotsb\subsetneqq\overline{\mathcal{O}(p_k)}\subsetneqq\dotsb$.

Moreover, the isometric GIET can be suspended to a smooth foliation, without singularities, on a 2-manifold. The exceptional (respectively dense) orbits of the GIET give rise to exceptional (respectively dense) leaves of the foliation. Finite genus 2-manifolds cannot support orientable foliations with the considered dynamics.

Let f be a diffeomorphism of a manifold preserving a Borel probability measure $\mu$ having no zero Lyapunov exponents and $\mathcal{W}^s$ a set with asymptotically attracting properties. An upper bound on the Hausdorff dimension of $\mathcal{W}^s$ is given in terms of the Lyapunov exponents and the measure-theoretic entropy. For a surface diffeomorphism, in particular, this upper bound relates, roughly speaking, to the dimension of $\mu$ in the direction of the unstable subspace. Furthermore, we apply this result to establish the equivalence of several properties of $\mu$: its statistical properties, its physical observability and its dimension-theoretical properties (Corollary 2.7).

Let (X, T) be a topological dynamical system and let $\mu$ be a T-invariant probability measure on X. In this paper, we study two properties of the notions of measure theoretical entropy for a measurable cover $\mathcal{U},\ h_\mu^+(\mathcal{U},T)$ and $h_\mu^-(\mathcal{U},T)$ introduced by P. P. Romagnoli (Ergod. Th. & Dynam. Sys. 23 (2003), 1601–1610). The main result of the paper states that entropy pairs for the measure $\mu$ can be defined using either $h_\mu^+$ or $h_\mu^-$. We also prove that both $h_\mu^+$ and $h_\mu^-$ have an ergodic decomposition and we use it to prove a local Abramov formula for $h_\mu^-$.

The $\lambda$-entropy of a right-resolving $\lambda$-graph system is introduced as the upper asymptotic growth rate of the number of vertices at level n. It is shown that shift equivalent right-resolving $\lambda$-graph systems have the same $\lambda$-entropy. The $\lambda$-entropies of right-resolving $\lambda$-graph systems that are invariantly associated to subshifts are invariants of topological conjugacy. Finite directed graphs are used in the construction of compact forward separated Shannon graphs that have the structure of a pushdown automaton and that are invariantly associated to the subshift they present. To these compact forward separated Shannon graphs there correspond $\lambda$-graph systems whose $\lambda$-entropy is equal to the topological entropy of the edge shift of the finite directed graph. Examples are given.

Cocyclic subshifts arise as the supports of matrix cocycles over a full shift and generalize topological Markov chains and sofic systems. We compute the topological entropy of a cocyclic subshift as the logarithm of the spectral radius of an appropriate transfer operator and give a concrete description of the measure of maximal entropy in terms of the eigenvectors. Unlike in the Markov or sofic case, the operator is infinite-dimensional and the entropy may be a logarithm of a transcendental number.

A diffeomorphism f of a compact manifold M is called almost Axiom-A if it is hyperbolic in a neighborhood of some compact f-invariant set $\Omega$, except in some singular set of neutral points. We prove that if there exists some f-invariant set of hyperbolic points with positive unstable-Lebesgue measure and such that for every point in this set the stable and unstable leaves are ‘long enough’, then f admits either a probability Sinai–Ruelle–Bowen (SRB)-measure or a $\sigma$-finite SRB-measure.

Krieger's embedding theorem for $\mathbb{Z}$ mixing shifts for finite type (SFTs) is extended to the $\mathbb{Z}^2$ case. We prove that if X is a non-periodic subshift ($\sigma^vx=x\Rightarrow v=0\in\mathbb{Z}^2$) and Y is a $\mathbb{Z}^2$ square-filling mixing SFT, then there exists a homomorphism $X\to Y$. The proof is a construction which begins by constructing Voronoi tilings using techniques from Part I (S. Lightwood. Morphisms from non-periodic $\mathbb{Z}^2$ subshifts I: constructing embeddings from homomorphisms. Ergod. Th. & Dynam. Sys.23 (2003), 587–609.). A tiling by Delaunay polygons is derived from Voronoi tiling. The union of the boundaries of the Delaunay polygons (referred to as the Delaunay graph) is itself tiled by trees. Words painted on the thickened trees combine to form words on the thickened infinite Delaunay graph. It is the point and sole purpose of square-filling that words on such thickened infinite graphs will correspond to points in the target space.

Combined with the results from Part I, this gives us the $\mathbb{Z}^2$ extension of Krieger's Embedding Theorem: if X is a non-periodic subshift and Y is a $\mathbb{Z}^2$ square-filling mixing SFT, then there exists an embedding $X\hookrightarrow Y$ if and only if $h(X)<h(Y)$, where h denotes the $\mathbb{Z}^2$ entropy. The techniques developed here play a central role in the proof of an embedding theorem for general $\mathbb{Z}^2$ subshifts into square-filling mixing SFTs which will be carried out in a subsequent paper.

In this paper, we prove a relation between robust transitivity and robust ergodicity for conservative diffeomorphisms. In dimension two, robustly transitive diffeomorphisms are robustly ergodic. In higher dimensions, we work with partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. We define almost robust ergodicity and prove that in the three-dimensional case robustly transitive diffeomorphisms are approximated by almost robustly ergodic ones. In higher dimensions under the condition of the central Lyapunov exponents being of the same sign, we prove that robustly transitive diffeomorphisms are robustly ergodic.