Let (X, S), (Y, T) be topological dynamical systems and \pi : X \rightarrow Y a factor map. A function F \in C(X) is a compensation function if P (F + \phi \circ \pi) = P (\phi) for all \phi \in C(Y). We present an example of a factor map \pi : X \rightarrow Y between subshifts of finite type X, Y that does not have a saturated compensation function and an example of a non-Markovian factor map with a saturated compensation function. Also we provide a necessary and sufficient condition for a certain type of factor map to have a saturated compensation function.

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.

We compute the homology of the integral manifolds of the restricted three-body problem—planar and spatial, unregularized and regularized. Holding the Jacobi constant fixed defines a three-dimensional algebraic set in the planar case and a five dimensional algebraic set in the spatial case (the integral manifolds). The singularities of the restricted problem due to collusions are removable, which defines the regularized problem.

There are five positive critical values of the Jacobi constant: one is due to a critical point at infinity, another is due to the Lagrangian critical points and three are due to the Eulerian critical points. The critical point at infinity occurs only in spatial problems. We compute the homology of the integral manifold for each regular value of the Jacobi constant. These computations show that at each critical value the integral manifolds undergo a bifurcation in their topology. The bifurcation due to a critical point at infinity shows that Birkhoff's conjecture is false even in the restricted problem.

Birkhoff also asked if the planar problem is the boundary of a cross section for the spatial problem. Our computations and homological criteria show that this can never happen in the restricted problem, but may be possible in the regularized problem for some values of the Jacobi constant. We also investigate the existence of global cross sections in each of the problems.

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

The unstable manifolds of hyperbolic systems admitting a one-rectangle Markov partition are here characterized up to homeomorphism and up to conjugacy of the underlying dynamics in a very easy-to-compute way. We associate to each one-rectangle system a word over the alphabet \{+,-\} describing the bending of the image rectangle with respect to the initial rectangle. We endow the set \{+,-\}^{\ast} of such words with a product \wedge having a dynamical origin. The structure of the non-commutative semigroup (\{+,-\}^{\ast},\wedge) is made completely explicit. The topological and dynamical properties of the unstable manifolds of one-rectangle systems are translated in terms of the decomposition of the associated words in such a semigroup.

We study from a combinatorial point of view quadratic fibered polynomials. The main object of this paper is the construction of a combinatorial model of a subset of the parameter space. This set (the configuration space) is proved to be a Hausdorff compact and connected space. Then, we prove that an abstract configuration which is non-recurrent can be uniquely realized by a quadratic fibered polynomial.

Let (X,T) be a Cantor minimal system associated with a properly ordered Bratteli diagram with the equal path number property. In this paper we will show that there exists a Toeplitz flow (Y,S) such that (X,T) and (Y,S) are strongly orbit equivalent. This is an affirmative answer to the open problem in R. Gjerde and Ø. Johansen (Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Th. & Dynam. Sys.20 (2000), 1687–1710).

Anderson and Putnam have recently shown that the space of all tilings of a substitution tiling scheme is a special case of the present author's ‘expanding attractor’. This resurrects an earlier conjecture of the present author in this special case: The tiling space of a substitution tiling space is a fiber bundle over the torus, with Cantor set fiber. If true, this could be important in the classification of these spaces, just as in the one-dimensional case. These spaces were known (in these two disciplines) to be locally the product of a disk and a Cantor set. Farrell and Jones gave an example showing our more inclusive conjecture is false. Here we show that the Penrose, Ammann and ‘stepped plane’ tiling spaces are in fact such bundles. Among dynamicists, the stepped plane tiling space (in special cases) is known as the ‘DA’ attractor of Smale. The formalism of substitution systems provides a Markov partition, and this is certainly part of their intuitive appeal. However, in dynamics such partitions are rare in dimensions greater than 2 and do not occur for the DA attractor. We consider a weaker concept, that omits this requirement. Finally, we construct an example like the Farrell–Jones example, in sufficient detail that it can be visualized as a smooth tiling space, as opposed to the more rigid geometric tiling space.