Stable accessibility of partially hyperbolic systems is central to their stable ergodicity, and we establish its $C^1$ -density among partially hyperbolic flows, as well as in the categories of volume-preserving, symplectic, and contact partially hyperbolic flows. As applications, we obtain on one hand in each of these four categories of flows the $C^1$ -density of the $C^1$ -stable topological transitivity and triviality of the centralizer, and on the other hand the $C^1$ -density of the $C^1$ -stable K-property of the natural volume in the latter three categories.

Given an $\unicode[STIX]{x1D6FC}>1$ and a $\unicode[STIX]{x1D703}$ with unbounded continued fraction entries, we characterize new relations between Sturmian subshifts with slope $\unicode[STIX]{x1D703}$ with respect to (i) an $\unicode[STIX]{x1D6FC}$-Hölder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $\unicode[STIX]{x1D703}$, and (iii) complexity notions which we call $\unicode[STIX]{x1D6FC}$-repetitiveness, $\unicode[STIX]{x1D6FC}$-repulsiveness and $\unicode[STIX]{x1D6FC}$-finiteness—generalizations of the properties known as linear repetitiveness, repulsiveness and power freeness, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is $2/(\unicode[STIX]{x1D6FC}+1)$.

We prove a normal form for strong magnetic fields on a closed, oriented surface and use it to derive two dynamical results for the associated flow. First, we show the existence of invariant tori and trapping regions provided a natural non-resonance condition holds. Second, we prove that the flow cannot be Zoll unless (i) the Riemannian metric has constant curvature and the magnetic function is constant, or (ii) the magnetic function vanishes and the metric is Zoll. We complement the second result by exhibiting an exotic magnetic field on a flat two-torus yielding a Zoll flow for arbitrarily weak rescalings.

It is known that if Ω is a Lebesgue space, T:Ω→Ω is a loosely Bernoulli transformation, and L is a fixed non-zero integer, then the transformation S = TL will again be loosely Bernoulli on each ergodic component. In this note, the above stated result is extended to include the case where L is an arbitrary integrable integer-valued function on Ω.

We study infinite systems of mean field weakly coupled intermittent maps in the Pomeau–Manneville scenario. We prove that the coupled system admits a unique ‘physical’ stationary state, to which all absolutely continuous states converge. Moreover, we show that suitably regular states converge polynomially.

We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge \exp (|z|^\alpha )$ for some $\alpha>0$ and all z outside a set of finite Lebesgue measure.

For typical cocycles over subshifts of finite type, we show that for any given orbit segment, we can construct a periodic orbit such that it shadows the given orbit segment and that the product of the cocycle along its orbit is a proximal linear map. Using this result, we show that suitable assumptions on the periodic orbits have consequences over the entire subshift.

We show that a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.

An extension from Riemann to Lorentz manifolds is proved for Anosov's theorem on the hyperbolicity (or exponential sensitivity to initial conditions) of the geodesic flow.

We find generalized conformal measures and equilibrium states for random dynamics generated by Ruelle expanding maps, under which the dynamics exhibits exponential decay of correlations. This extends results by Baladi [Correlation spectrum of quenched and annealed equilibrium states for random expanding maps. Comm. Math. Phys. 186 (1997), 671–700] and Carvalho et al [Semigroup actions of expanding maps. J. Stat. Phys. 116(1) (2017), 114–136], where the randomness is driven by an independent and identically distributed process and the phase space is assumed to be compact. We give applications in the context of weighted non-autonomous iterated function systems, free semigroup actions and introduce a boundary of equilibria for not necessarily free semigroup actions.

Based on previous work of the authors, to any S-adic development of a subshift X a ‘directive sequence’ of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S-adic development. The issuing rich picture enables one to deduce results about X with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result, we also exhibit, for any integer $d \geq 2$, an S-adic development of a minimal, aperiodic, uniquely ergodic subshift X, where all level alphabets $\mathcal A_n$ have cardinality $d,$ while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subseteq \mathcal A_n^{\mathbb {Z}}$.

We study the problem of lifting ergodicity to certain skew-product extensions over shifts of finite type. This can be done by computing the asymptotic range of the defining cocycle. For a class of ergodic shift-invariant measures and a class of functions (defining conservative extensions) we give a characterization of ergodicity in terms of weights of periodic orbits in the shift space.

We consider the McMullen maps $f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$ with $\unicode{x3bb} \in \mathbb {C}^{*}$ and $n \geq 3$. We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and Sierpiński holes) are quasi-circles with Hausdorff dimension strictly between $1$ and $2$.

We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Cesàro means of non-expansive maps on Banach spaces.

We prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.

Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.

Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.

Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$, namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$. We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

We prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

For any primitive substitution whose Perron eigenvalue is a Pisot unit, we construct a domain exchange that is measurably conjugate to the subshift. Additionally, we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitutions, we show that the subshift is either a finite extension of a torus translation or its eigenvalues are roots of unity. Furthermore, we provide an algorithm to compute eigenvalues of the subshift associated with any primitive pseudo-unimodular substitution.

We look at constructions of aperiodic subshifts of finite type (SFTs) on fundamental groups of graph of groups. In particular, we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb {F}_n\times \mathbb {Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb {Z}^2$ inside an SFT on $\mathbb {F}_n\times \mathbb {Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$. In the case of $\mathbb {F}_n\times \mathbb {Z}$, the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.