A fundamental question in the field of cohomology of dynamical systems is to determine when there are solutions to the coboundary equation:

$$ \begin{align*} f = g - g \circ T. \end{align*} $$

In many cases, T is given to be an ergodic invertible measure-preserving transformation on a standard probability space $(X, {\mathcal B}, \mu )$ and is contained in $L^p$ for $p \geq 0$ . We extend previous results by showing for any measurable f that is non-zero on a set of positive measure, the class of measure-preserving T with a measurable solution g is meager (including the case where $\int _X f\,d\mu = 0$ ). From this fact, a natural question arises: given f, does there always exist a solution pair T and g? In regards to this question, our main results are as follows. Given measurable f, there exist an ergodic invertible measure-preserving transformation T and measurable function g such that $f(x) = g(x) - g(Tx)$ for almost every (a.e.) $x\in X$ , if and only if $\int _{f> 0} f\,d\mu = - \int _{f < 0} f\,d\mu $ (whether finite or $\infty $ ). Given mean-zero $f \in L^p(\mu )$ for $p \geq 1$ , there exist an ergodic invertible measure-preserving T and $g \in L^{p-1}(\mu )$ such that $f(x) = g(x) - g( Tx )$ for a.e. $x \in X$ . In some sense, the previous existence result is the best possible. For $p \geq 1$ , there exists a dense $G_{\delta }$ set of mean-zero $f \in L^p(\mu )$ such that for any ergodic invertible measure-preserving T and any measurable g such that $f(x) = g(x) - g(Tx)$ almost everywhere, then $g \notin L^q(\mu )$ for $q> p - 1$ . Finally, it is shown that we cannot expect finite moments for solutions g, when $f \in L^1(\mu )$ . In particular, given any such that $\lim _{x\to \infty } \phi (x) = \infty $ , there exist mean-zero $f \in L^1(\mu )$ such that for any solutions T and g, the transfer function g satisfies:

$$ \begin{align*} \int_{X} \phi ( | g(x) | )\,d\mu = \infty. \end{align*} $$

For a $C^{1+\alpha }$ diffeomorphism f of a compact smooth manifold, we give a necessary and sufficient condition that guarantees that if the set of hyperbolic Lyapunov–Perron regular points has positive volume, then f preserves a smooth measure. We use recent results on symbolic coding of $\chi $ -non-uniformly hyperbolic sets and results concerning the existence of SRB measures for them.

An iterative square root of a self-map f is a self-map g such that $g(g(\cdot ))=f(\cdot )$ . We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. They are used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^{m}$ and to the whole of $\mathbb {R}^{m}$ for every positive integer $m.$ However, we also prove that every continuous self-map on a space homeomorphic to the unit cube in $\mathbb {R}^{m}$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.

Let $(\mathbb {D}^2,\mathscr {F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb {D}^2$ defined by the linear vector field

$$ \begin{align*} z \frac{\partial}{\partial z}+ \unicode{x3bb} w \frac{\partial}{\partial w}, \end{align*} $$

where $\unicode{x3bb} \in \mathbb {C}^*$ . Such a foliation has a non-degenerate singularity at the origin ${0:=(0,0) \in \mathbb {C}^2}$ . Let T be a harmonic current directed by $\mathscr {F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$ . Assume $T\neq 0$ . The Lelong number of T at $0$ describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when $\unicode{x3bb} \notin \mathbb {R}$ , that is, when $0$ is a hyperbolic singularity, the Lelong number at $0$ vanishes. Suppose the trivial extension $\tilde {T}$ across $0$ is $dd^c$ -closed. For the non-hyperbolic case $\unicode{x3bb} \in \mathbb {R}^*$ , we prove that the Lelong number at $0$ :

1. (1) is strictly positive if $\unicode{x3bb}>0$ ;

2. (2) vanishes if $\unicode{x3bb} \in \mathbb {Q}_{<0}$ ;

3. (3) vanishes if $\unicode{x3bb} <0$ and T is invariant under the action of some cofinite subgroup of the monodromy group.

In this paper, we introduce the notion of planar two-center Stark–Zeeman systems and define four $J^{+}$ -like invariants for their periodic orbits. The construction is based on a previous construction for a planar one-center Stark–Zeeman system in [K. Cieliebak, U. Frauenfelder and O. van Koert. Periodic orbits in the restricted three-body problem and Arnold’s $J^+$ -invariant. Regul. Chaotic Dyn. 22(4) (2017), 408–434] as well as Levi-Civita and Birkhoff regularizations. We analyze the relationship among these invariants and show that they are largely independent, based on a new construction called interior connected sum.

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f : \mathbb {N} \to \mathbb {N}$ with $f(n)/n$ increasing and $\sum 1/f(n) < \infty $ , that there exists an extremely elevated staircase with word complexity $p(n) = o(f(n))$ . This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

It is shown that each locally compact second countable non-(T) group G admits non-strongly ergodic weakly mixing IDPFT Poisson actions of any possible Krieger type. These actions are amenable if and only if G is amenable. If G has the Haagerup property, then (and only then) these actions can be chosen of 0-type. If G is amenable, then G admits weakly mixing Bernoulli actions of arbitrary Krieger type.

We present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math. 397 (2022), 52], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. & Dynam. Sys. 39 (2019), 3075–3088], which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math. 308 (2017), 599–626] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

We prove an extension of the Moore–Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a ‘conditional’ Pontryagin duality for spaces of abstract measurable maps.

Furstenberg–Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg–Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi.

For $k \geq 2$ , we prove that in a $C^{1}$ -open and $C^{k}$ -dense set of some classes of $C^{k}$ -Anosov flows, all Lyapunov exponents have multiplicity one with respect to appropriate measures. The classes are geodesic flows with equilibrium states of Holder-continuous potentials, volume-preserving flows, and all fiber-bunched Anosov flows with equilibrium states of Holder-continuous potentials.

We show that every isometric action on a Cantor set is conjugate to an inverse limit of actions on finite sets; and that every faithful isometric action by a finitely generated amenable group is residually finite.

We show that the complete positive entropy (CPE) class $\alpha $ of Barbieri and García-Ramos contains a one-dimensional subshift for all countable ordinals $\alpha $ , that is, the process of alternating topological and transitive closure on the entropy pairs relation of a subshift can end on an arbitrary ordinal. This is the composition of three constructions. We first realize every ordinal as the length of an abstract ‘close-up’ process on a countable compact space. Next, we realize any abstract process on a compact zero-dimensional metrizable space as the process started from a shift-invariant relation on a subshift, the crucial construction being the implementation of every compact metrizable zero-dimensional space as an open invariant quotient of a subshift. Finally, we realize any shift-invariant relation E on a subshift X as the entropy pair relation of a supershift $Y \supset X$ , and under strong technical assumptions, we can make the CPE process on Y end on the same ordinal as the close-up process of E.