We present a modified version of the well-known geometric Lorenz attractor. It consists of a $C^1$ open set ${\mathcal O}$ of vector fields in ${\mathbb R}^3$ having an attracting region ${\mathcal U}$ satisfying three properties. Namely, a unique singularity $\sigma $; a unique attractor $\Lambda $ including the singular point and the maximal invariant in ${\mathcal U}$ has at most two chain recurrence classes, which are $\Lambda $ and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along with the union of $2$ codimension $1$ submanifolds which split ${\mathcal O}$ into three regions. By crossing this collision locus, the attractor and the horseshoe may merge into a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expels the singular point $\sigma $ and becomes a horseshoe, and the horseshoe absorbs $\sigma $ becoming a Lorenz attractor.

We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by $T(x,y) = (E (x), C(x,y))$, where E is an expanding map of $\mathbb {T}^u$ and C is a contracting map on each fiber. If $\inf |\!\det DT| \inf \| (D_yC)^{-1}\| ^{-2s}>1$ for some ${s<r-(({u+d})/{2}+1)}$, $r \geq 2$, and T satisfies a transversality condition between overlaps of iterates of T (a condition which we prove to be $C^r$-generic under mild assumptions), then the SRB measure $\mu _T$ of T is absolutely continuous and its density $h_T$ belongs to the Sobolev space $H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$. When $s> {u}/{2}$, it is also valid that the density $h_T$ is differentiable with respect to T. Similar results are proved for thermodynamical quantities for potentials close to the geometric potential.

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,\ldots , f_k$ such that the functions $f_i\circ T^n$, $n\in {\mathbb {Z}}$, $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.

Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K-points of V. We introduce algebraic sofic subshifts ${\Sigma \subset A^G}$ and study endomorphisms $\tau \colon \Sigma \to \Sigma $. We generalize several results for dynamical invariant sets and nilpotency of $\tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau $ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and $\Sigma $ is topologically mixing, we show that $\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.

For $\mathscr {B} \subseteq \mathbb {N} $, the $ \mathscr {B} $-free subshift $ X_{\eta } $ is the orbit closure of the characteristic function of the set of $ \mathscr {B} $-free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\eta } $, have their analogues for $ X_{\eta } $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures [G. Keller. Generalized heredity in $\mathcal B$-free systems. Stoch. Dyn. 21(3) (2021), Paper No. 2140008]. A central assumption in our work is that $\eta ^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_{\eta } $) is regular. From this, we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_{\eta } $ from above and below.

For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.

Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$-dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $, which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets $\{\Lambda _n\}_{n\geq 1}$. The limit behaviour of the Carathéodory singular dimension of $\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.

Let $K\subset {\mathbb {R}}^d$ be a self-similar set generated by an iterated function system $\{\varphi _i\}_{i=1}^m$ satisfying the strong separation condition and let f be a contracting similitude with $f(K)\subseteq K$. We show that $f(K)$ is relatively open in K if all $\varphi _i$ share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question of Elekes, Keleti and Máthé [Ergod. Th. & Dynam. Sys. 30 (2010), 399–440]. As a byproduct of our argument, when $d=1$ and K admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction ratios of opposite signs, we show that K is symmetric. This partially answers a question of Feng and Wang [Adv. Math. 222 (2009), 1964–1981].

The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the non-integrability of the restricted three-body problem both in the planar and spatial cases for any non-zero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales–Ramis and Morales–Ramis–Simó theories. We emphasize that our results are not just applications of the theories.