We prove statistical stability for a family of Lorenz attractors with a $C^{1+\unicode[STIX]{x1D6FC}}$ stable foliation.

We generalize a result of Hochman in two simultaneous directions: instead of realizing an arbitrary effectively closed $\mathbb{Z}^{d}$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^{2}$. Let $H$ be a finitely generated group and $G=\mathbb{Z}^{2}\rtimes _{\unicode[STIX]{x1D711}}H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,T)$ where $Y\subset \{0,1\}^{\mathbb{N}}$, there exists a $G$-subshift of finite type $(X,\unicode[STIX]{x1D70E})$ such that the $H$-subaction of $(X,\unicode[STIX]{x1D70E})$ is an extension of $(Y,T)$. In the case where $T$ is an expansive action, a subshift conjugated to $(Y,T)$ can be obtained as the $H$-projective subdynamics of a sofic $G$-subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.

We study the action of the affine group $G$ of $\mathbb{R}^{d}$ on the space $X_{k,\,d}$ of $k$-dimensional affine subspaces. Given a compactly supported Zariski dense probability measure $\unicode[STIX]{x1D707}$ on $G$, we show that $X_{k,d}$ supports a $\unicode[STIX]{x1D707}$-stationary measure $\unicode[STIX]{x1D708}$ if and only if the $(k+1)\text{th}$ Lyapunov exponent of $\unicode[STIX]{x1D707}$ is strictly negative. In particular, when $\unicode[STIX]{x1D707}$ is symmetric, $\unicode[STIX]{x1D708}$ exists if and only if $2k\geq d$.

Thermodynamic formalism, the theory of equilibrium states, is studied both in dynamical systems and probability theory. Various closely related notions have been developed: e.g. Dobrushin–Lanford–Ruelle Gibbs, Bowen–Gibbs and $g$-measures. We discuss the relation between Gibbs and $g$-measures in a one-dimensional context. Often $g$-measures are also Gibbs, but recently an example to the contrary has been presented. In this paper we discuss exactly when a $g$-measure is Gibbs and how this relates to notions such as uniqueness and reversibility of $g$-measures.

We define a subgroup of the universal sofic group, obtained as the normalizer of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each sofic representation can be conjugated inside this subgroup.

We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT($-1$) metrics on $X$, which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.

It is shown that each conservative non-singular Bernoulli shift is either of type $\mathit{II}_{1}$ or $\mathit{III}_{1}$. Moreover, in the latter case the corresponding Maharam extension of the shift is a $K$-automorphism. This extends earlier results obtained by Kosloff for equilibrial shifts. Non-equilibrial shifts of type $\mathit{III}_{1}$ are constructed. We further generalize (partly) the main results to non-singular Markov shifts.

A linear subspace $E$ of $\mathbb{R}^{n}$ has colored local rules if there exists a finite set of decorated tiles whose tilings are digitizations of $E$. The local rules are weak if the digitizations can slightly wander around $E$. We prove that a linear subspace has weak colored local rules if and only if it is computable. This goes beyond previous results, all based on algebraic subspaces. We prove an analogous characterization for sets of linear subspaces, including the set of all the linear subspaces of $\mathbb{R}^{n}$.

Polynomial Abel differential equations are considered a model problem for the classical Poincaré center–focus problem for planar polynomial systems of ordinary differential equations. In the last few decades, several works pointed out that all centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.

We consider homeomorphisms $f,h$ generating a faithful $\mathit{BS}(1,n)$-action on a closed surface $S$, that is, $hfh^{-1}=f^{n}$ for some $n\geq 2$. According to Guelman and Liousse [Actions of Baumslag–Solitar groups on surfaces. Discrete Contin. Dyn. Syst. A 5 (2013), 1945–1964], after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\unicode[STIX]{x1D6EC}$ of the action, included in $\text{Fix}(f)$. Here, we suppose that $f$ and $h$ are $C^{1}$ in a neighborhood of $\unicode[STIX]{x1D6EC}$ and any point $x\in \unicode[STIX]{x1D6EC}$ admits an $h$-unstable manifold $W^{u}(x)$. Using Bonatti’s techniques, we prove that either there exists an integer $N$ such that $W^{u}(x)$ is included in $\text{Fix}(f^{N})$ or there is a lower bound for the norm of the differential of $h$ depending only on $n$ and the Riemannian metric on $S$. Combining the last statement with a result of Alonso, Guelman and Xavier [Actions of solvable Baumslag–Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete Contin. Dyn. Syst. to appear], we show that any faithful action of $\mathit{BS}(1,n)$ on $S$ with $h$ a pseudo-Anosov homeomorphism has a finite orbit containing singularities of $h$; moreover, if $f$ is isotopic to the identity, it is entirely contained in the singular set of $h$. As a consequence, there is no faithful $C^{1}$-action of $\mathit{BS}(1,n)$ on the torus with $h$ Anosov.

In this article pattern statistics of typical cubical cut and project sets are studied. We give estimates for the rate of convergence of appearances of patches to their asymptotic frequencies. We also give bounds for repetitivity and repulsivity functions. The proofs use ideas and tools developed in discrepancy theory.

We prove several results for the arithmetic dynamics of monomial maps, including Silverman’s conjectures on height growth, dynamical Mordell–Lang conjecture, and dynamical Manin–Mumford conjecture. These results were originally known for monomial maps on algebraic tori. We extend them to arbitrary toric varieties.

We consider suspension flows over uniquely ergodic skew-translations on a $d$-dimensional torus $\mathbb{T}^{d}$ for $d\geq 2$. We prove that there exists a set $\mathscr{R}$ of smooth functions, which is dense in the space $\mathscr{C}(\mathbb{T}^{d})$ of continuous functions, such that every roof function in $\mathscr{R}$ which is not cohomologous to a constant induces a mixing suspension flow. We also construct a dense set of mixing examples which is explicitly described in terms of their Fourier coefficients. In the language of nilflows on nilmanifolds, our result implies that, for every uniquely ergodic nilflow on a quasi-abelian filiform nilmanifold, there exists a dense subspace of smooth time-changes in which mixing occurs if and only if the time-change is not cohomologous to a constant. This generalizes a theorem by Avila, Forni and Ulcigrai [Mixing for time-changes of Heisenberg nilflows. J. Differential Geom.89(3) (2011), 369–410] for the classical Heisenberg group.

We first consider the dynamics of a class of meromorphic skew products having superattracting fixed points or fixed indeterminacy points at the origin. Our theorem asserts that, if a map has a suitable weight, then it is conjugate to the associated monomial map on an invariant open set whose closure contains the origin. We next extend this result to a wider class of meromorphic maps such that the eigenvalues of the associated matrices are real and greater than $1$.