Let $\{h_{a,b} : a,b \in \mathbb{R}, b \neq 0\}$ denote the Hénon family of quadratic polynomial diffeomorphisms of $\mathbb{R}^2$, with $b$ equal to the Jacobian of $h_{a,b}$. In this paper, we describe the locus of parameter values $(a,b)$ such that $0 < |b| < 0.06$, and the restriction of $h_{a,b}$ to its non-wandering set is topologically conjugate to the horseshoe map. The boundary of the horseshoe locus is shown to be characterized by a homoclinic tangency which is part of a generic unfolding.

We consider actions of the free semigroup with two generators on the real line, where the generators act as affine maps, one contracting and one expanding, with distinct fixed points. Then every orbit is dense in a half-line, which leads to the question whether it is, in some sense, uniformly distributed. We present answers to this question for various interpretations of the phrase ‘uniformly distributed’.

We show that a periodic orbit of large period of a diffeomorphism or flow either admits a dominated splitting of a prescribed strength or can be turned into a sink or a source by a $C^1$-small perturbation along the orbit. As a consequence we show that the linear Poincaré flow of a $C^1$-vector field admits a dominated splitting over any robustly transitive set.

We study transversely Lorentzian foliations of closed 3-manifolds. We classify them under a completeness hypothesis and deduce the dual classification of codimension-one geodesically complete timelike totally geodesic foliations. Also we provide an example of a Lorentzian foliation of a compact 3-manifold which is equicontinuous on a proper dense open subset of the manifold, and hence not transversely complete.

In this paper we study analytic properties of compensator and Dulac expansions in a single variable. A Dulac expansion is formed by monomial terms that may contain a specific logarithmic factor. In compensator expansions this logarithmic factor is deformed. We first consider Dulac expansions when the power of the logarithm is either 0 or 1. Here we construct an explicit exponential scaling in the space of coefficients which in an exponentially narrow horn, up to rescaling and division, leads to a polynomial expansion. A similar result holds for the compensator case. This result is applied to the bifurcation theory of limit cycles in planar vector fields. The setting consists of families that unfold a given Hamiltonian in a dissipative way. This leading part is of Morse type, which leads to the following three cases. The first concerns a Hamiltonian function that is regular on an annulus, and the second a Hamiltonian function with a non-degenerate minimum defined on a disc. In the third case the Hamiltonian function has a non-degenerate saddle point with a saddle connection. The first case, by an appropriate scaling, recovers the generic theory of the saddle node of limit cycles and its cuspoid degeneracies, while the second case similarly recovers the generic theory of limit cycles subordinate to the codimension $k$ Hopf bifurcations $k = 1,2,\dotsc$. The third case enables a novel study of generic bifurcations of limit cycles subordinate to homoclinic bifurcations. We then describe how the above analytic result is applied to bifurcations of limit cycles. For appropriate one-dimensional Poincaré maps, the fixed points correspond to the limit cycles. The fixed point sets (or zero-sets of the associated displacement functions) are studied by contact equivalence singularity theory. The cases where the Hamiltonian is defined on an annulus or a disc can be reduced directly to catastrophe theory. In the third case, the displacement functions are known to have compensator expansions, whose first approximations are Dulac expansions. Application of our analytic result implies that, in an exponentially narrow horn near a homoclinic loop, the bifucation theory of limit cycles again reduces to catastrophe theory.

We prove that for a suitable (open) class of open, smoothly bounded domains in the cotangent bundle of a surface of genus $g \geq 2$ any exact symplectomorphism is homotopic to one which is smooth up to the boundary. In particular, such boundaries are not unseen in the sense of Eliashberg–Hofer. The contact boundaries of these domains have Anosov Reeb flows. The methods employed include the properties of such flows in three dimensions, and analysis of two-dimensional transverse flow maps along the lines of Thurston, and thus appear restricted to the case of three-dimensional contact boundaries.

In this paper we show that, for every Choquet simplex $K$ and for every $d>1$, there exists a ${\mathbb Z}^d$-Toeplitz system whose set of invariant probability measures is affine homeomorphic to $K$. Then, we conclude that $K$ may be realized as the set of invariant probability measures of a tiling system $(\Omega_T,{\mathbb R}^d)$.

We consider locally compact groups $G$ admitting a topologically transitive $\mathbb{Z}^d$-action by automorphisms. It is shown that such a group $G$ has a compact normal subgroup $K$ of $G$, invariant under the action, such that $G/K$ is a product of (finitely many) locally compact fields of characteristic zero; moreover, the totally disconnected fields in the decomposition can be chosen to be invariant under the $\mathbb{Z}^d$-action and such that the $\mathbb{Z}^d$-action is via scalar multiplication by non-zero elements of the field. Under the additional conditions that $G$ be finite dimensional and ‘locally finitely generated’ we conclude that $K$ as above is connected and contained in the center of $G$. We describe some examples to point out the significance of the conditions involved.

For each $n>1$, we construct explicitly a rigid weakly mixing rank-$n$ transformation with homogeneous spectrum of multiplicity $n$. The existence of such transformations was established recently by O. Ageev via Baire category arguments (a new short category proof is also given here). As an application, for any subset $M\subset\mathbb N$ containing 1, a weakly mixing transformation whose essential range for the spectral multiplicity equals $n\cdot M$ is constructed.

This paper addresses the following topics relating to the structure of hyperbolic sets: first, hyperbolic sets that are not contained in locally maximal hyperbolic sets; second, the existence of a Markov partition for a hyperbolic set. We construct new examples of hyperbolic sets which are not contained in locally maximal hyperbolic sets. The first example is robust under perturbations and can be constructed on any compact manifold of dimension greater than one. The second example is robust, topologically transitive, and constructed on a four-dimensional manifold. The third example is volume-preserving and constructed on $\mathbb{R}^4$. We show that every hyperbolic set is included in a hyperbolic set with a Markov partition. In addition, we describe a condition that ensures a hyperbolic set is included in a locally maximal hyperbolic set.

Suppose $M$ is a compact surface and $\Lambda\subset M$ is a non-trivial mixing hyperbolic attractor for some $f\in\mathrm{Diff}(M)$. We show that if $\Lambda$ is a hyperbolic set for some $g\in\mathrm{Diff}(M)$, then $\Lambda$ is a non-trivial mixing hyperbolic attractor or repeller for $g$.

We give a non-commutative generalization of classical symbolic coding in the presence of a synchronizing word. This is done by a scattering theoretical approach. Classically, the existence of a synchronizing word turns out to be equivalent to asymptotic completeness of the corresponding Markov process. A criterion for asymptotic completeness in general is provided by the regularity of an associated extended transition operator. Commutative and non-commutative examples are analysed.

A dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta{\mathbb{N}}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, Glasner (On tame enveloping semigroups. Colloq. Math.105 (2006), 283–395) calls the system tame. In this paper, we study the tame system under an Abelian group action. We introduce the notion of scrambled pairs for a dynamical system under an Abelian group action and show that a tame system has no scrambled pair. At the same time, we give some sufficient conditions such that a pair is a scrambled one. Moreover, using these sufficient conditions we prove that a minimal tame system under an Abelian group action is almost automorphic and uniquely ergodic. This gives a positive answer to Problem 2.5 in the paper cited above. Finally, we prove that for ${\mathbb Z}$-actions a tame system is $M$-null, that is, the metric sequence entropy is zero for any invariant measure and any increasing sequence of natural numbers. We also give examples to show that tameness is strictly weaker than nullness for ${\mathbb Z}$-actions.

We prove that a polynomial vector field on $\mathbb{C}^n$, $n\geq 2$, whose corresponding projective foliation has only singularities of hyperbolic type is linear in some affine chart provided that it is transverse to a sequence of spheres bounding balls exhausting ${\mathbb{C}}^n$.

We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital $C^*$-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Let $A$ be a stably finite simple unital $C^*$-algebra, and let $\alpha$ be an automorphism of $A$ which has the tracial Rokhlin property. Suppose $A$ has real rank zero and stable rank one, and suppose that the order on projections over $A$ is determined by traces. Then the crossed product algebra $C^* (\mathbb{Z}, A, \alpha)$ also has these three properties. We also present examples of $C^*$-algebras $A$ with automorphisms $\alpha$ which satisfy the above assumptions, but such that $C^* (\mathbb{Z}, A, \alpha)$ does not have tracial rank zero.

We introduce a general class of automorphisms of rotation algebras, the non-commutative Furstenberg transformations. We prove that fully irrational non-commutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by traces (Blackadar's Second Fundamental Comparability Question). We also prove that several classes of simple quotients of the $C$*-algebras of discrete subgroups of five-dimensional nilpotent Lie groups, considered by Milnes and Walters, are crossed products of simple $C$*-algebras ($C$*-algebras of minimal ordinary Furstenberg transformations) by automorphisms which have the tracial Rokhlin property. It follows that these algebras also have stable rank one, real rank zero, and order on projections determined by traces.

We show that the volume entropy of surfaces with unit systole is bounded from above by a constant which does not depend on the metric, which answers a question raised by A. Katok. This upper bound follows from a comparison between the geometric and algebraic lengths on the fundamental groups of surfaces.