We discuss the dynamics of a class of non-ergodic piecewise affine maps of the torus. These maps exhibit highly complex and little understood behavior. We present computer graphics of some examples and analyses of some with a decreasing degree of completeness. For the best understood example, we show that the torus splits into three invariant sets on which the dynamics are quite different. These are: the orbit of the discontinuity set, the complement of this set in its closure, and the complement of the closure. There are still some unsolved problems concerning the orbit of the discontinuity set. However we do know that there are intervals of periodic orbits and at least one infinite orbit. The map on the second invariant set is minimal and uniquely ergodic. The third invariant set is one of full Lebesgue measure and consists of a countable number of open octagons whose points are periodic. Their orbits can be described in terms of a symbolism obtained from an equal length substitution rule or the triadic odometer.

Soient f un endomorphisme holomorphe de \mathbb{C}\mathsf{P}^k de degré au moins 2 et \mu sa mesure d'équilibre. Nous montrons que l'extension naturelle de (f,\mu) possède la propriété de Bernoulli.

We study exactness and maximal automorphic factors of C^3 unimodal maps of the interval. We show that for a large class of infinitely renormalizable maps, the maximal automorphic factor is an odometer with an ergodic non-singular measure. We give conditions under which maps with absorbing Cantor sets have an irrational rotation on a circle as a maximal automorphic factor, as well as giving exact examples of this type. We also prove that every C^3 S-unimodal map with no attractor is exact with respect to Lebesgue measure. Additional results about measurable attractors in locally compact metric spaces are given.

Given a piecewise invertible map T:X\to X and a weight g:X\rightarrow\ ]0,\infty[, a conformal measure \nu is a probability measure on X such that, for all measurable A\subset X with T:A\to TA invertible, \nu(TA)= \lambda \int_{A}\frac{1}{g}\ d\nu with a constant \lambda>0. Such a measure is an essential tool for the study of equilibrium states. Assuming that the topological pressure of the boundary is small, that \log g has bounded distortion and an irreducibility condition, we build such a conformal measure.

Residual entropy of a topological system is defined as the infimum increase of entropy necessary to build a symbolic extension of this system. If no symbolic extension exists then residual entropy is set at infinity. In this paper we provide a direct formula for the residual entropy of a system on a totally disconnected compact space in terms of basic notions of conditional entropies viewed as functions of invariant measures. This formula allows us to evaluate residual entropy in many examples as well as to construct new examples with arbitrarily preset topological and residual entropies. The appendix contains a condition equivalent to asymptotic h-expansiveness.

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and \Gamma a lattice in G. We study automorphic forms for \Gamma if G is of real rank one with some additional assumptions, using a dynamical approach based on properties of the homogeneous flow on \Gamma\backslash G and a Livshitz type theorem we prove for such a flow. In the Hermitian case G=SU(n,1) we construct relative Poincaré series associated to closed geodesics on \Gamma\backslash G/K for one-dimensional representations of K, and prove that they span the corresponding spaces of holomorphic cusp forms.

We show that for a smooth contact Anosov flow on a closed three manifold the measure of maximal entropy is in the Lebesgue class if and only if the flow is, up to finite covers, conjugate to the geodesic flow of a metric of constant negative curvature on a closed surface. This shows that the ratio between the measure theoretic entropy and the topological entropy of a contact Anosov flow is strictly smaller than one on any closed three manifold which is not a Seifert bundle.

There are four Bowen–Franks groups associated to each shift of finite type. For an irreducible shift of finite type, we show that a 4-tuple of automorphisms corresponding to the four Bowen–Franks groups can be induced simultaneously by a specific path of flow equivalence from the shift to itself, if and only if it is F-compatible. The F-compatibleness defined in this paper describes completely the intrinsic relations among the four automorphisms induced by a flow equivalence. This result is one of the key ingredients in classifying reducible shifts of finite type up to flow equivalence. In the mean time, it also discloses a new and sharp difference between the invariants of flow equivalence for an irreducible shift of finite type, and the invariants of stable isomorphism for the associated simple Cuntz–Krieger algebra.

We consider the problem of expressing the regularity of different objects which appear in Anosov systems in terms of Sobolev spaces. We show in examples that one cannot expect any improvements in the regularity of Anosov splitting from the classical results that give regularity in W^{r_0,\infty}. We also develop a Sobolev regularity theory for solutions of cohomology equations, both commutative and non-commutative, and the bootstrap of regularity arguments from Sobolev to other classes for a wide variety of functional equations involving Anosov systems.

Given an isolated fixed point p of index one, of an area- and orientation-preserving homeomorphism f of a surface, we define a type (positive, negative) for a lift of f to the universal cover of D\setminus \{p\}, where D is a small disc neighbourhood of p. Using this concept, we establish a new kind of fixed point theorem. Furthermore, a refinement of the Arnold conjecture for a surface homeomorphism is obtained.

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.

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.

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.

In [1], Lemma 6.6 is wrong in the case d_U=d_V=2. The proof of this lemma includes an uncorrectable error. Thus, the proof of Theorem 2 is invalid.

Using the same notation as used in [S], we give a corrected proof of Lemma 1.10 in [S].