Given a factor code from a shift of finite type onto a sofic shift , the class degree of is defined to be the minimal number of transition classes over the points of . In this paper, we investigate the structure of transition classes and present several dynamical properties analogous to the properties of fibers of finite-to-one factor codes. As a corollary, we show that for an irreducible factor triple, there cannot be a transition between two distinct transition classes over a right transitive point, answering a question raised by Quas.

Consider the space of sequences of  letters ordered lexicographically. We study the set  of all maximal sequences for which the asymptotic proportions  of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of is called the -infimax sequence, or the -minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the -infimax for every in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.

Let be a Hausdorff, étale groupoid that is minimal and topologically principal. We show that is purely infinite simple if and only if all the non-zero positive elements of are infinite in . If is a Hausdorff, ample groupoid, then we show that is purely infinite simple if and only if every non-zero projection in is infinite in . We then show how this result applies to -graph -algebras. Finally, we investigate strongly purely infinite groupoid -algebras.

In a recent paper the authors introduced universal and exotic generalized fixed-point algebras for weakly proper group actions on -algebras. Here we extend the notion of weakly proper actions to actions on Hilbert modules. As a result we obtain several imprimitivity theorems establishing important Morita equivalences between universal, reduced, or exotic crossed products and appropriate universal, reduced, or exotic fixed-point algebras, respectively. In particular, we obtain an exotic version of Green’s imprimitivity theorem and a very general version of the symmetric imprimitivity theorem by weakly proper actions of product groups . In addition, we study functorial properties of generalized fixed-point algebras for equivariant categories of -algebras based on correspondences.

We show that a germ of a holomorphic one-dimensional foliation at a singularity in a space of dimension two admits a holomorphic first integral if and only if there are infinitely many closed leaves and a finite number of separatrices, with each separatrix having linearizable holonomy. Indeed, if there are infinitely many closed leaves and the set of separatrices is finite, then the foliation admits either a holomorphic first integral or a formal simple integrating factor of Darboux type.

According to the work of Dennis Sullivan [A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 5–14], there exists a smooth flow on the five-sphere all of whose orbits are periodic, although there is no uniform bound on their periods. The question addressed in this article is whether these type of examples can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact but there is no uniform bound for their volumes? We develop tools to attack the previous question and show that it has a negative answer provided that all periodic leaves have finite holonomy.

In this paper we characterize measure-theoretical eigenvalues of Toeplitz Bratteli–Vershik minimal systems of finite topological rank which are not associated to a continuous eigenfunction. Several examples are provided to illustrate the different situations that can occur.

We construct an increasing sequence of natural numbers with the property that is dense in for any , and a continuous measure on the circle such that . Moreover, for every fixed , the set is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.

We consider the dynamics on the -algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani [KMS states for gauge action on . Math. Japon.29 (1984), 607–619] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo–Martin–Schwinger (KMS) state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of examples. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph.

In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain assumptions the Birkhoff spectrum is real analytic. We also show that new phenomena occur; indeed, the spectrum can be constant or it can have points where it is not analytic. Conditions for these to happen are obtained. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.

Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal. Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the latter property must have zero entropy.

An algebraic flip system is an action of the infinite dihedral group by automorphisms of a compact abelian group . In this paper, a fundamental structure theorem is established for irreducible algebraic flip systems, that is, systems for which the only closed invariant subgroups of are finite. Using irreducible systems as a foundation, for expansive algebraic flip systems, periodic point counting estimates are obtained that lead to the orbit growth estimate

Let be the attractor of a linear iterated function system (IFS) , on the real line satisfying the generalized finite type condition (whose invariant open set is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math.208 (2007), 647–671]as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let be the dimension of . In this paper, we state that

We consider Hölder continuous fiber bunched -valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of a diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question of whether an Anosov diffeomorphism is smoothly conjugate to a -small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of , for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.