We prove that there is a residual subset $\mathcal{I}$ of ${\rm Diff}^1({\it M})$ such that any homoclinic class of a diffeomorphism $f\in \mathcal{I}$ having saddles of indices $\alpha$ and $\beta$ contains a dense subset of saddles of index $\tau$ for every $\tau\in [\alpha,\beta]\cap \mathbb{N}$. We also derive some consequences from this result about the Lyapunov exponents of periodic points and the sort of bifurcations inside homoclinic classes of $C^1$-generic diffeomorphisms.

We show that some very naturally occurring energy manifolds that are induced by second-order Lagrangians $L =L(u,u',u'')$ are not, in general, of contact type in $(\mathbb{R}^4,\omega)$. We also comment on the more general question whether there exist any contact forms on these energy manifolds for which the associated Reeb vector field coincides with the Hamiltonian vector field.

On the cylinder ${\mathbb T}\times{\mathbb R}^1$, smooth dynamical systems are constructed that, in spite of being smoothly conjugate on an invariant covering, are not topologically conjugate globally.

In this paper we compute the Hausdorff dimension of the set $D(\varphi_n)$ of points on divergent trajectories of the homogeneous flow $\varphi_n$ induced by the one-parameter subgroup $\mathop{\rm diag}(e^t,e^{-t})$ acting by left multiplication on the product space $G^n/\Gamma^n$, where $G=\mathop{\rm SL}(2,{\mathbb R})$ and $\Gamma=\mathop{\rm SL}(2,{\mathbb Z})$. We prove that $\dim_H D(\varphi_n)=3n-\frac{1}{2}$ for $n\ge2$.

Let $T$ be a finite subset of the complex unit circle ${\mathbb{S}^1}$ and define $f: {\mathbb{S}^1} \mapsto {\mathbb{S}^1}$ by $f(z) = z^d$. Let $\mathrm{CH}(T)$ denote the convex hull of $T$. If $\mathrm{card}(T) = N \geq 3$, then $\mathrm{CH}(T)$ defines a polygon with $N$ sides. The $N$-gon $\mathrm{CH}(T)$ is called a wandering N-gon if for every two non-negative integers $i \neq j,\ \mathrm{CH}(f^i(T))$ and $\mathrm{CH}(f^j(T))$ are disjoint $N$-gons. A non-degenerate chord of ${\mathbb{S}^1}$ is said to be critical if its two endpoints have the same image under $f$. Then for a critical chord, it is natural to define its (forward) orbit by the forward iterates of the endpoints. Similarly, we call a critical chord recurrent if one of its endpoints is recurrent under $f$. The main result of our study is that a wandering $N$-gon has at least $N-1$ recurrent critical chords in its limit set (defined in a natural way) having pairwise disjoint, infinite orbits. Using this result, we are able to strengthen results of Blokh, Kiwi, Levin and Thurston about wandering polygons of laminations. We also discuss some applications to the dynamics of polynomials. In particular, our study implies that if $v$ is a wandering non-precritical vertex of a locally connected polynomial Julia set, then there exists at least $\mathrm{ord}(v) -1$ recurrent critical points with pairwise disjoint orbits, all having the same $\omega$-limit set as $v$. Thus, we likewise strengthen results about wandering vertices of polynomial Julia sets.

We show that any probability preserving transformation of a metric space is mixing as soon as there are no non-constant $L^2$-functions which are invariant under both the stable and unstable distributions. This generalizes a result of Babillot.

A reverse weak type inequality for the conjugate function is generalized to the ergodic case. The inequality allows us to prove the Riesz–Zygmund theorem (if $f\geq 0$ and $f,\widetilde{f}\in L$, then $f\in L\log L$) for the ergodic Hilbert transform.

We study a skew product with a curve of neutral points. We show that there exists a unique absolutely continuous invariant probability measure, and that the Birkhoff averages of a sufficiently smooth observable converge to a normal law or a stable law, depending on the average of the observable along the neutral curve.

In this paper we prove that there exist bounded orbit injections from minimal $\mathbb{Z}^{2}$ actions of a Cantor set $T$ and $S$ into a common action $R$ if and only if the suspension spaces associated to $T$ and $S$ are homeomorphic. In this way we prove a two-dimensional analog of a result of Parry and Sullivan on flow equivalence and discrete cross-sections for minimal systems. At the same time the result is a topological analog of a result of del Junco and Rudolph on Kakutani equivalence for ergodic $\mathbb{Z}^{d}$ actions. We also prove a structural result about such suspension spaces. Namely, that they are a finite union of products of Cantor sets with polygons, $C_{i}\times P_{i}$, after an identification on the boundary, $C_{i}\times \partial P_{i}$, with the action given by ${\mathbb{R}}^{2}$ on the polygon. The polygons $P_{i}$ can be chosen to have properties associated with Voronoi or Delaunay tilings corresponding to a set of points located uniformly throughout the plane.

In this paper we complete the analysis of the stability of periodic points of piecewise isometric systems that are defined in the entire Euclidean space. We focus on the case where the determinant of the matrix ${\rm Id}-R_{w}$ is zero, where $R_{w}$ is the linear part of the return map generated by the periodic point whose coding is made of a countable repetition of the block $w$. The case where the determinant of ${\rm Id}-R_{w}$ is non-zero was studied in Mendes and Nicol (Periodicity and recurrence in piecewise rotations of Euclidean spaces. Int. J. Bif. Chaos, 14(7), 2353–2361) but still the statement is included here for the sake of completeness.

Let ${\mathcal{A}}^{{\mathbb{Z}}^D}$ be the Cantor space of ${\mathbb{Z}}^D$-indexed configurations in a finite alphabet ${\mathcal{A}}$, and let $\sigma$ be the ${\mathbb{Z}}^D$-action of shifts on ${\mathcal{A}}^{{\mathbb{Z}}^D}$. A cellular automaton is a continuous, $\sigma$-commuting self-map $\Phi$ of ${\mathcal{A}}^{{\mathbb{Z}}^D}$, and a $\Phi$-invariant subshift is a closed, $(\Phi,\sigma)$-invariant subset ${\mathfrak{A}}\subset{\mathcal{A}}^{{\mathbb{Z}}^D}$. Suppose that ${\mathbf{a}}\in{\mathcal{A}}^{{\mathbb{Z}}^D}$ is ${\mathfrak{A}}$-admissible everywhere except for some small region that we call a defect. It has been empirically observed that such defects persist under iteration of $\Phi$, and often propagate like ‘particles’ which coalesce or annihilate on contact. We construct algebraic invariants for these defects, which explain their persistence under $\Phi$, and partly explain the outcomes of their collisions. Some invariants are based on the cocycles of multidimensional subshifts; others arise from the higher-dimensional (co)homology/homotopy groups for subshifts, obtained by generalizing the Conway–Lagarias tiling groups and the Geller–Propp fundamental group.

We prove that if $G$ is a countable, discrete group having infinite normal subgroups with the relative property (T) of Kazhdan–Margulis, then the Bernoulli shift action of $G$ on $\prod_{g \in G} (X_0, \mu_0)_g$, for $(X_{0},\mu_{0})$ an arbitrary non-trivial probability space, has first cohomology group isomorphic to the character group of $G$.

Alves, Bonatti and Viana have recently shown that physical (or Sinai–Ruelle–Bowen) invariant measures exist for partially hyperbolic diffeomorphisms with mostly expanding center-unstable direction. In this paper we prove that such systems are statistically stable, that is, nearby diffeomorphisms have nearby physical measures.

This paper is concerned with Borel 1-cocycles on ergodic flows on standard Borel measure spaces and a certain type of group coactions on (separable) factors.