Lyapunov graphs carry dynamical information of gradient-like flows as well as topological information of their phase space which is taken to be a closed orientable n-manifold. In this paper we will show that an abstract Lyapunov graph $L(h_0, \dotsc,h_n,\kappa)$ in dimension n greater than 2, with cycle number $\kappa$, satisfies the Poincaré–Hopf inequalities if and only if it satisfies the Morse inequalities and the first Betti number, $\gamma_1$, is greater than or equal to $\kappa$. We also show a continuation theorem for abstract Lyapunov graphs with the presence of cycles. Finally, a family of Lyapunov graphs $\mathcal{L}(h_0, \dotsc, h_n,\kappa)$ with fixed pre-assigned data $(h_0, \dotsc, h_n,\kappa)$ is associated with the Morse polytope, $\mathcal{P}_{\kappa}(h_0,\dotsc, h_n)$, determined by the Morse inequalities for the given data.

We prove that a minimal self-mapping of a compact 2-manifold has tree-like fibers (i.e. all points have preimages which are connected, at most one-dimensional and with trivial shape). We also prove that the only 2-manifolds (compact or not) which admit minimal maps are either finite unions of tori, or finite unions of Klein bottles.

We study a d-dimensional coupled map lattice consisting of hyperbolic toral automorphisms (Arnold cat maps) that are weakly coupled by an analytic map. We construct the Sinai–Ruelle–Bowen measure for this system and study its marginals on the tori. We prove that they are absolutely continuous with respect to Lebesgue measure if and only if the coupling satisfies a non-degeneracy condition.

In this paper we obtain (almost) optimal results concerning randomly weighted one-sided ergodic Hilbert transforms. Given an iid sequence of centered random variables (Xn) in L log L, we show that there exists a universal set $\Omega'$ of probability 1 such that for any $\omega\in \Omega'$ the realization $(X_n(\omega))$ is good for the almost everywhere convergence of the weighted one-sided ergodic Hilbert transform associated with any dynamical system and any $g\in L\log L$. The method applies to powers along subsequences with ‘small’ growth and when considering Dunford–Schwartz operators instead of pointwise transformation. If the (Xn) are symmetric, but only in L log log L, we obtain a slightly weaker result.

I would like to mention here several improvements or additional results to the paper ‘On randomly weighted one-sided ergodic Hilbert transforms’. We obtained there that iid random weights in L log L are universally good for the almost sure convergence of the weighted one-sided ergodic Hilbert transform for any Dunford–Schwartz operator in a probability space $(X,\Sigma, \mu)$ and any function g satisfying $\int_X |g|{\rm log}^+|g| \,d\mu<+\infty$.

We characterize when a lower entropy locally compact subshift S embeds into a locally compact mixing Markov shift T. As in the compact case (when T is a shift of finite type) the existence of an embedding depends on S only through its periodic orbit counts. However, in contrast to the compact case, the topology of the target system T becomes important. This is demonstrated with the help of the Zeta Function Lemma, which in particular characterizes the periodic orbit counts and entropies of locally compact mixing Markov shifts.

We consider families of one-dimensional maps on the circle which mix at sub-polynomial rates. Such maps will have an indifferent fixed point, and we show that the rate of mixing of these maps is determined by the precise degeneracy of the fixed point. By constructing suitable induced maps with a countable Markov partition and applying a probabilistic coupling argument we are able to give good estimates on the rate of mixing for these families of maps. The underlying question we pose is the following: given any specific rate of mixing can we construct a map which has that rate of mixing? For a large class of specific rates we explain how to construct the appropriate map.

We define a notion of the Haagerup property for measure-preserving standard equivalence relations. Given such a relation R on X with finite invariant measure $\mu$, we prove that R has the Haagerup property if and only if the associated finite von Neumann algebra L(R) (see J. Feldman and C. C. Moore. Ergodic equivalence relations, cohomology and von Neumann algebras II. Trans. Amer. Math. Soc.234 (1977), 325–350) has relative property H in the sense of Popa with respect to its natural Cartan subalgebra $L^{\infty}(X,\mu)$. We also prove that if G is a countable group such that R = RG has the Haagerup property and if R is ergodic, then G cannot have Kazhdan's property T.

We investigate rigidity of measurable structure for higher-rank Abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a zero entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.

We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This implies that the ergodic averages of a continuous function on X along a polynomial sequence of translations on X converge pointwise.

Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of $\mathbb{Z}^{d}$ by translations on X is uniformly distributed on the union of several sub-nilmanifolds of X. As a corollary we obtain the pointwise ergodic theorem for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold.

We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a two-symbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing.

We consider the class of $\mathbb{R}^n$ extensions of Anosov diffeomorphisms on infranilmanifolds, and find necessary and sufficient conditions for topological transitivity. In particular, if the fiber is $\mathbb{R}$, the existence of a semi-orbit with the projection on $\mathbb{R}$ unbounded from above and from below is equivalent to topological transitivity. We also show that in the above class topological transitivity and stable topological transitivity are equivalent.

Let $f: M \to M$ be a diffeomorphism defined in a three-dimensional compact boundary-less manifold M. We prove that for an open dense set, C1-robustly expansive homoclinic classes H(p) for f are hyperbolic. A diffeomorphism f is $\alpha$-expansive on a compact invariant set K if there is $\alpha>0$ such that for all $x,y\in K$, if ${\rm dist}(f^n(x),f^n(y))\leq \alpha$ for all $n\in \mathbb Z$ then x = y. By ‘robustly’ we mean that there is $\alpha>0$ such that for all nearby diffeomorphisms g, the homoclinic class H(pg) of the continuation of p is $\alpha$-expansive.

We use a notion (introduced in Topology41 (2002), 1119–1212), which is stronger than the concept of filtration pair, to prove a stable/unstable manifold general theorem for local homeomorphisms of the plane in a neighborhood of an isolated fixed point.