We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties.

The homogenizability of linear transport equations with periodic data is investigated. It can be interpreted in terms of the dynamical system properties of the associated ODEs. De Giorgi conjectured that all equations of this kind are homogenizable. In this paper, the homogenizability is proved is the two-dimensional case for non-vanishing functions (partial results in this direction have been previously proved by other authors). On the other hand, in the three-dimensional case, an example of a non-homogenizable equation is given.

In this paper we introduce two notions of measure-theoretical entropy for covers and one notion of topological entropy for open covers. These notions extend the classical concept of measure-theoretical entropy of measurable partitions to measurable covers. Using these notions we prove a local variational principle for any given open cover.

For a continuous transformation f of a compact metric space (X,d) and any continuous function \phi on X we consider sets of the form

K_{\alpha} =\bigg\{x\in X:\lim_{n\to\infty} \frac 1n \sum_{i=0}^{n-1} \phi( f^i(x))=\alpha \bigg\},\quad\alpha\in\R.

For transformations satisfying the specification property we prove the following Variational Principle

h_{\rm top}(f,K_{\alpha}) = \sup\bigg( h_\mu(f): \mu\text{ is invariant and } \int\phi \,d\mu=\alpha \bigg),

where h_{\rm top}(f,\cdot) is the topological entropy of non-compact sets. Using this result we are able to obtain a complete description of the multifractal spectrum for Lyapunov exponents of the so-called Manneville–Pomeau map, which is an interval map with an indifferent fixed point.

We also consider multi-dimensional multifractal spectra and establish a contraction principle.

We construct dynamical partitions of the torus for a diffeomorphism that is isotopic to the identity. The existence and the combinatorics of the partitions is solely determined by the rotation set of the diffeomorphism. When the rotation set consists of a single non-resonant vector, there is a whole hierarchy of partitions analogous to the partitions of the circle into the closest return intervals under an irrational circle rotation. In particular, all such torus maps are infinitely renormalizable in a natural sense.

In this paper we consider random dynamical systems generated by compositions of one-sided independent and identically distributed random diffeomorphisms of class C2 on a compact manifold. We prove an entropy formula for such random dynamical systems without assuming the SRB condition. This result is the random version of the main result obtained by Ledrappier and Young (The metric entropy of diffeomorphisms. Part II: Relations between entropy, exponents and dimension. Ann. Math. 122 (1985), 540–574).

In this paper we study the behavior of diffeomorphisms, contained in the closure \overline{\mathcal{A}_\alpha} (in the inductive limit topology) of the set \mathcal{A}_\alpha of real-analytic diffeomorphisms of the torus \mathbb T^2, which are conjugated to the rotation R_\alpha:(x,y)\mapsto (x + \alpha , y) by an analytic measure-preserving transformation. We show that for a generic \alpha\in [0,1], \overline{\mathcal{A}_\alpha} contains a dense set of uniquely ergodic diffeomorphisms. We also prove that \overline{\mathcal{A}_\alpha} contains a dense set of diffeomorphisms that are minimal and non-ergodic.

Let $\Sigma = (X, \sigma)$ be a topological dynamical system where $\sigma$ is a homeomorphism in a compact metric space X. Denote by $A(\Sigma)$ the transformation group C*-algebra associated with this system. We describe the shrinking steps of the non-wandering set $\Omega(\sigma)$ down to the Birkhoff center (depth of the center) in terms of a composition series of the particular ideal of type 1 in $A(\Sigma)$, which corresponds to the center. The result implies C*-algebraic characterizations of the cases where the depth is 0 and 1. We also give the structure of dynamical systems for which the associated C*-algebras become algebras with continuous traces.

Let f be a real rational function with all critical points on the extended real axis and of even order. Then:

(1) f carries no invariant line field on the Julia set unless it is doubly covered by an integral torus endomorphism (a Lattés example); and

(2) f|J(f) has only finitely many ergodic components.

In this paper we introduce foundational techniques and prove the following: if X is a \mathbb{Z}^d subshift without periodic points, if Y is a \mathbb{Z}^d square mixing subshift of finite type containing a finite orbit and if there exists a homomorphism X\rightarrow Y, then X embeds into Y if and only if h(X)<h(Y). For the proof, clopen markers are used to generate Voronoi tiles whose thickened boundaries are coded using the homomorphism. The entropy gap and the square mixing permit the construction of an injective code on the tile interiors. A second paper will show that \mathbb{Z}^2 square filling mixing shifts of finite type are square mixing and that homomorphisms exist, resulting in an extension of Krieger's Embedding Theorem to \mathbb{Z}^2 subshifts.

We draw attention to the following mistakes which appear in the above-mentioned paper (pp. 732–734). We are grateful to J.-F. Mattei who pointed out with an example that the notion of singular holonomy should be enlarged and consequently the Structure Theorem restated. The main result (Theorem 1) remains untouched. We refer to the same notation and terminology of the paper.

The topological transitivity of non-compact group extensions of topologically mixing subshifts of finite type has been studied recently by Niţică. We build on these methods, and give the first examples of stably transitive non-compact group extensions of hyperbolic dynamical systems. Our examples include extensions of hyperbolic basic sets by the Euclidean group SE(n) for n even, n\ge4.

We show that if f is a C^2 diffeomorphism with positive entropy on an n-dimensional closed manifold (n \geq 2) then f is chaotic in the sense of Li–Yorke, and moreover f is \ast-chaotic on the closure of the stable manifold for some point. The notion of ‘\ast-chaos’ was introduced by Kato in relation to Li–Yorke's chaos in the context of topological dynamics.

This paper investigates the set of angles of the parameter rays which land on the real slice [-2,1/4] of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as bi-accessible points of quadratic Julia sets are given.

In the second paragraph on page 30 [1], we wrote ‘Making a critical point precritical decreases the number of points about which we have to worry’. Indeed, we stopped worrying about these points so much, that we forgot to mention them in several statements. This includes Lemma 3.13, Theorem 3.14 and the Main Theorem. Thus, the possibility of critical points being precritical (not only sinking or recurrent) has to be added to those statements. In particular, the Main Theorem should read as follows.

We prove that a subshift (X,T) is linearly recurrent if and only if it is a primitive and proper S-adic subshift. This corrects Proposition 6 in F. Durand (Ergod. Th. &: Dynam. Sys.20 (2000), 1061–1078).