We introduce and study the notion of hereditary frequent hypercyclicity, which is a reinforcement of the well-known concept of frequent hypercyclicity. This notion is useful for the study of the dynamical properties of direct sums of operators; in particular, a basic observation is that the direct sum of a hereditarily frequently hypercyclic operator with any frequently hypercyclic operator is frequently hypercyclic. Among other results, we show that operators satisfying the frequent hypercyclicity criterion are hereditarily frequently hypercyclic, as well as a large class of operators whose unimodular eigenvectors are spanning with respect to the Lebesgue measure. However, we exhibit two frequently hypercyclic weighted shifts $B_w,B_{w'}$ on $c_0(\mathbb {Z}_+)$ whose direct sum ${B_w\oplus B_{w'}}$ is not $\mathcal {U}$-frequently hypercyclic (so that neither of them is hereditarily frequently hypercyclic), and we construct a C-type operator on $\ell _p(\mathbb {Z}_+)$, $1\le p<\infty $, which is frequently hypercyclic but not hereditarily frequently hypercyclic. We also solve several problems concerning disjoint frequent hypercyclicity: we show that for every $N\in \mathbb {N}$, any disjoint frequently hypercyclic N-tuple of operators $(T_1,\ldots ,T_N)$ can be extended to a disjoint frequently hypercyclic $(N+1)$-tuple $(T_1,\ldots ,T_N, T_{N+1})$ as soon as the underlying space supports a hereditarily frequently hypercyclic operator; we construct a disjoint frequently hypercyclic pair which is not densely disjoint hypercyclic; and we show that the pair $(D,\tau _a)$ is disjoint frequently hypercyclic, where D is the derivation operator acting on the space of entire functions and $\tau _a$ is the operator of translation by $a\in \mathbb {C}\setminus \{ 0\}$. Part of our results are in fact obtained in the general setting of Furstenberg families.

Using a perturbation result established by Galatolo and Lucena [Spectral gap and quantitative statistical stability for systems with contracting fibers and Lorenz-like maps. Discrete Contin. Dyn. Syst. 40(3) (2020), 1309–1360], we obtain quantitative estimates on the continuity of the invariant densities and entropies of the physical measures for some families of piecewise expanding maps. We apply these results to a family of two-dimensional tent maps.

We prove that every locally compact second countable group G arises as the outer automorphism group $\operatorname{Out} M$ of a II1 factor, which was so far only known for totally disconnected groups, compact groups, and a few isolated examples. We obtain this result by proving that every locally compact second countable group is a centralizer group, a class of Polish groups that arise naturally in ergodic theory and that may all be realized as $\operatorname{Out} M$.

We prove several results showing that every locally finite Borel graph whose large-scale geometry is ‘tree-like’ induces a treeable equivalence relation. In particular, our hypotheses hold if each component of the original graph either has bounded tree-width or is quasi-isometric to a tree, answering a question of Tucker-Drob. In the latter case, we moreover show that there exists a Borel quasi-isometry to a Borel forest, under the additional assumption of (componentwise) bounded degree. We also extend these results on quasi-treeings to Borel proper metric spaces. In fact, our most general result shows treeability of countable Borel equivalence relations equipped with an abstract wallspace structure on each class obeying some local finiteness conditions, which we call a proper walling. The proof is based on the Stone duality between proper wallings and median graphs (i.e., CAT(0) cube complexes). Finally, we strengthen the conclusion of treeability in these results to hyperfiniteness in the case where the original graph has one (selected) end per component, generalizing the same result for trees due to Dougherty–Jackson–Kechris.

Starting from a uniquely ergodic action of a locally compact group G on a compact space $X_0$, we consider non-commutative skew-product extensions of the dynamics, on the crossed product $C(X_0)\rtimes _\alpha {\mathbb Z}$, through a $1$-cocycle of G in ${\mathbb T}$, with $\alpha $ commuting with the given dynamics. We first prove that any two such skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity with respect to the fixed-point subalgebra by characterizing both in terms of the cocycle assigning the dynamics. The set of all invariant states is also determined: it is affinely homeomorphic with ${\mathcal P}({\mathbb T})$, the Borel probability measures on the one-dimensional torus ${\mathbb T}$, as long as the system is not uniquely ergodic. Finally, we show that unique ergodicity with respect to the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra.

Jones [‘Two subfactors and the algebraic decomposition of bimodules over $II_1$ factors’, Acta Math. Vietnam 33(3) (2008), 209–218] proposed the study of ‘two subfactors’ of a $II_1$ factor as a quantization of two closed subspaces in a Hilbert space. Motivated by this, we initiate a systematic study of a special class of two subfactors, namely a pair of spin model subfactors. We characterize which pairs of distinct complex Hadamard matrices in $M_n(\mathbb {C})$ give rise to distinct spin model subfactors. Then, a detailed investigation is carried out for $n=2$, where the spin model subfactors correspond to $\mathbb {Z}_2$-actions on the hyperfinite type $II_1$ factor R. We observe that the intersection of the pair of spin model subfactors in this case is a nonirreducible vertex model subfactor and we characterize it as a diagonal subfactor. A few key invariants for the pair of spin model subfactors are computed to understand their relative positions.

We study linear random walks on the torus and show a quantitative equidistribution statement, under the assumption that the Zariski closure of the acting group is semisimple.

We develop combinatorial tools to study partial rigidity within the class of minimal $\mathcal {S}$-adic subshifts. By leveraging the combinatorial data of well-chosen Kakutani–Rokhlin partitions, we establish a necessary and sufficient condition for partial rigidity. Additionally, we provide an explicit expression to compute the partial rigidity rate and an associated partial rigidity sequence. As applications, we compute the partial rigidity rate for a variety of constant length substitution subshifts, such as the Thue–Morse subshift, where we determine a partial rigidity rate of 2/3. We also exhibit non-rigid substitution subshifts with partial rigidity rates arbitrarily close to 1 and, as a consequence, using products of the aforementioned substitutions, we obtain that any number in $[0, 1]$ is the partial rigidity rate of a system.

This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in Mierczyński et al. [18]. Two different approaches to its existence are presented. The state space X will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$, and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains an exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal.

We prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$, can be non-trivially and isometrically embedded in a family of piecewise isometries. In particular, this proves the existence of invariant curves for piecewise isometries, reminiscent of Kolmogorov–Arnold–Moser (KAM) curves for area-preserving maps, which are not unions of circle arcs or line segments.

We prove a full measurable version of Vizing’s theorem for bounded degree Borel graphs, that is, we show that every Borel graph $\mathcal {G}$ of degree uniformly bounded by $\Delta \in \mathbb {N}$ defined on a standard probability space $(X,\mu )$ admits a $\mu $-measurable proper edge coloring with $(\Delta +1)$-many colors. This answers a question of Marks [Question 4.9, J. Amer. Math. Soc. 29 (2016)] also stated in Kechris and Marks as a part of [Problem 6.13, survey (2020)], and extends the result of the author and Pikhurko [Adv. Math. 374, (2020)], who derived the same conclusion under the additional assumption that the measure $\mu $ is $\mathcal {G}$-invariant.

In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group G from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$, for some infinite sequence $(b_n)_{n\in \mathbb {N}}$ and some $t\in G$. In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in [13].

Let G be a compact group, let $\mathcal {B}$ be a unital C$^*$-algebra, and let $(\mathcal {A},G,\alpha )$ be a free C$^*$-dynamical system, in the sense of Ellwood, with fixed point algebra $\mathcal {B}$. We prove that $(\mathcal {A},G,\alpha )$ can be realized as the G-continuous part of the invariants of an equivariant coaction of G on a corner of $\mathcal {B} \otimes {\mathcal {K}}({\mathfrak {H}})$ for a certain Hilbert space ${\mathfrak {H}}$ that arises from the freeness of the action. This extends a result by Wassermann for free and ergodic C$^*$-dynamical systems. As an application, we show that any faithful $^*$-representation of $\mathcal {B}$ on a Hilbert space ${\mathfrak {H}}_{\mathcal {B}}$ gives rise to a faithful covariant representation of $(\mathcal {A},G,\alpha )$ on some truncation of ${\mathfrak {H}}_{\mathcal {B}} \otimes {\mathfrak {H}}$.

A cylindrical cascade on $\mathbb {T}^d\times \mathbb {R}^r$ can be seen as a deterministic random walk on $\mathbb {R}^r$ driven by an observable over the irrational toral translation on the base torus. We prove that, when the observable is the indicator function of a generic (straight) rectangle in $\mathbb {T}^2$, the cascade on $\mathbb {T}^2\times \mathbb {R}$ is ergodic for a $G_{\delta }$-dense set of translation vectors. We also provide examples of ergodic cylindrical cascades in higher dimensions with more restrictive conditions on the side lengths of the rectangles.

We study the class $\operatorname {Erg}^\perp $ of automorphisms which are disjoint with all ergodic systems. We prove that the identities are the only multipliers of $\operatorname {Erg}^\perp ,$ that is, each automorphism whose every joining with an element of $\operatorname {Erg}^{\perp }$ yields a system which is again an element of $\operatorname {Erg}^{\perp }$, must be an identity. Despite this fact, we show that $\operatorname {Erg}^\perp $ is closed by taking Cartesian products. Finally, we prove that there are non-identity elements in $\operatorname {Erg}^\perp $ whose self-joinings always yield elements in $\operatorname {Erg}^\perp $. This shows that there are non-trivial characteristic classes included in $\operatorname {Erg}^\perp $.

Let $(X,\mathcal {B},\mu ,T)$ be a probability-preserving system with X compact and T a homeomorphism. We show that if every point in $X\times X$ is two-sided recurrent, then $h_{\mu }(T)=0$, resolving a problem of Benjamin Weiss, and that if $h_{\mu }(T)=\infty $, then every full-measure set in X contains mean-asymptotic pairs (that is, the associated process is not tight), resolving a problem of Ornstein and Weiss.

We build a Shannon orbit equivalence between the universal odometer and a variety of rank-one systems. This is done in a unified manner using what we call flexible classes of rank-one transformations. Our main result is that every flexible class contains an element which is Shannon orbit equivalent to the universal odometer. Since a typical example of flexible class is $\{T\}$ when T is an odometer, our work generalizes a recent result by Kerr and Li, stating that every odometer is Shannon orbit equivalent to the universal odometer. When the flexible class is a singleton, the rank-one transformation given by the main result is explicit. This applies to odometers and Chacon’s map. We also prove that strongly mixing systems, systems with a given eigenvalue, or irrational rotations whose angle belongs to any fixed non-empty open subset of the real line form flexible classes. In particular, strong mixing, rationality or irrationality of the eigenvalues are not preserved under Shannon orbit equivalence.

Here we consider the discrete time dynamics described by a transformation $T:M \to M$, where T is either the action of shift $T=\sigma$ on the symbolic space $M=\{1,2, \ldots,d\}^{\mathbb{N}}$, or, T describes the action of a d to 1 expanding transformation $T:S^1 \to S^1$ of class $C^{1+\alpha}$ (for example $x \to T(x) =\mathrm{d} x $ (mod 1)), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of Hölder equilibrium probabilities is an analytical manifold and carries a natural Riemannian metric. Given a certain normalized Hölder potential A denote by $\mu_A \in \mathcal{N}$ the associated equilibrium probability. The set of tangent vectors X (functions $X: M \to \mathbb{R}$) to the manifold $\mathcal{N}$ at the point µA (a subspace of the Hilbert space $L^2(\mu_A)$) coincides with the kernel of the Ruelle operator for the normalized potential A. The Riemannian norm $|X|=|X|_A$ of the vector X, which is tangent to $\mathcal{N}$ at the point µA, is described via the asymptotic variance, that is, satisfies

$ |X|^2 = \langle X, X \rangle = \lim_{n \to \infty}\frac{1}{n} \int (\sum_{i=0}^{n-1} X\circ T^i )^2 \,\mathrm{d} \mu_A$.

Consider an orthonormal basis Xi, $i \in \mathbb{N}$, for the tangent space at µA. For any two orthonormal vectors X and Y on the basis, the curvature $K(X,Y)$ is

\begin{equation*}K(X,Y) = \frac{1}{4}[ \sum_{i=1}^\infty (\int X Y X_i \,\mathrm{d} \mu_A)^2 - \sum_{i=1}^\infty \int X^2 X_i \,\mathrm{d} \mu_A \int Y^2 X_i \,\mathrm{d} \mu_A ].\end{equation*}

When the equilibrium probabilities µA is the set of invariant Markov probabilities on $\{0,1\}^{\mathbb{N}}\subset \mathcal{N}$, introducing an orthonormal basis $\hat{a}_y$, indexed by finite words y, we show explicit expressions for $K(\hat{a}_x,\hat{a}_z)$, which is a finite sum. These values can be positive or negative depending on A and the words x and z. Words $x,z$ with large length can eventually produce large negative curvature $K(\hat{a}_x,\hat{a}_z)$. If $x, z$ do not begin with the same letter, then $K(\hat{a}_x,\hat{a}_z)=0$.

In this paper, we study ergodic $\mathbb {Z}^r$-actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions, there are always directions which expand significantly a given measurable set with positive measure. Among other things, we use this result to prove that the set of volumes of all r-simplices with vertices in a set with positive upper density must contain an infinite arithmetic progression, thus showing a discrete density analogue of a classical result by Graham.

When implementing Markov Chain Monte Carlo (MCMC) algorithms, perturbation caused by numerical errors is sometimes inevitable. This paper studies how the perturbation of MCMC affects the convergence speed and approximation accuracy. Our results show that when the original Markov chain converges to stationarity fast enough and the perturbed transition kernel is a good approximation to the original transition kernel, the corresponding perturbed sampler has fast convergence speed and high approximation accuracy as well. Our convergence analysis is conducted under either the Wasserstein metric or the $\chi^2$ metric, both are widely used in the literature. The results can be extended to obtain non-asymptotic error bounds for MCMC estimators. We demonstrate how to apply our convergence and approximation results to the analysis of specific sampling algorithms, including Random walk Metropolis, Metropolis adjusted Langevin algorithm with perturbed target densities, and parallel tempering Monte Carlo with perturbed densities. Finally, we present some simple numerical examples to verify our theoretical claims.