It is well known that the dynamical spectrum of an ergodic measure dynamical system is related to the diffraction measure of a typical element of the system. This situation includes ergodic subshifts from symbolic dynamics as well as ergodic Delone dynamical systems, both via suitable embeddings. The connection is rather well understood when the spectrum is pure point, where the two spectral notions are essentially equivalent. In general, however, the dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of finite local complexity and establish the equivalence of the dynamical spectrum with a collection of diffraction spectra of the system and certain factors. This equivalence gives access to the dynamical spectrum via these diffraction spectra. It is particularly useful as the diffraction spectra are often simpler to determine and, in many cases, only very few of them need to be calculated.

We show that, under the assumption of chain transitivity, the shadowing property is equivalent to the thick shadowing property. We also show that, if ${\mathcal{F}}$ is a family with the Ramsey property, then an arbitrary sequence of points in a chain transitive space can be ${\it\varepsilon}$-shadowed (for any ${\it\varepsilon}$) on a set in ${\mathcal{F}}$.

We prove a continuation result for Morse decompositions under tubular singular semiflow perturbations, which generalizes a corresponding result from Carbinatto and Rybakowski [Morse decompositions in the absence of uniqueness, II. Topol. Methods Nonlinear Anal.22 (2003), 15–51] and is applicable to cases in which the phase space of the perturbed semiflow is not necessarily homeomorphic to a product of metric spaces having as a factor the phase space of the limiting semiflow. We apply this result to singularly perturbed second-order differential equations on differential manifolds.

In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, as well as the corresponding question of equivalence of associated Cuntz algebra ${\mathcal{O}}_{N}$-representations. We do this by associating certain monic representations of ${\mathcal{O}}_{N}$ to quasi-stationary Markov measures and then proving that equivalence for a pair of measures is decided by unitary equivalence of the corresponding pair of representations.

In this paper we prove the following rigidity result: let ${\it\varphi}$ be a $C^{\infty }$ topologically mixing transversely symplectic Anosov flow. If (i) its weak stable and weak unstable distributions are $C^{\infty }$ and (ii) its Hamenstädt metrics are sub-Riemannian, then up to finite covers and a constant change of time scale, ${\it\varphi}$ is $C^{\infty }$ flow conjugate to the geodesic flow of a closed locally symmetric Riemannian space of rank one.

In the context of the Franks–Misiurewicz conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterize these maps by the existence of an invariant ‘foliation’ by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalizing a well-known result of Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

We define directional recurrence for infinite measure preserving $\mathbb{Z}^{d}$ actions both intrinsically and via the unit suspension flow and prove that the two definitions are equivalent. We study the structure of the set of recurrent directions and show it is always a $G_{{\it\delta}}$ set. We construct an example of a recurrent action with no recurrent directions, answering a question posed in a 2007 paper of Daniel J. Rudolph. We also show by example that it is possible for a recurrent action to not be recurrent in an irrational direction even if all its sub-actions are recurrent.

On a compact real interval, the spaces of all transitive maps, all piecewise monotone transitive maps and all piecewise linear transitive maps are considered with the uniform metric. It is proved that they are contractible and uniformly locally arcwise connected. Then the spaces of all piecewise monotone transitive maps with given number of pieces as well as various unions of such spaces are considered and their connectedness properties are studied.

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family

$$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$

We study the long-term behavior of injective piecewise contractions of the interval. We prove that every injective piecewise contraction with $n-1$ discontinuities has at most $n$ periodic orbits and is topologically conjugate to a piecewise linear contraction.

We establish a necessary and sufficient condition for a Poisson suspension to be prime. The proof is based on the Fock space structure of the $L^{2}$-space of the Poisson suspension. We give examples of explicit infinite measure-preserving systems, in particular of non-singular compact group rotations that give rise to prime Poisson suspensions. We also compare some properties of so far known prime transformations with those of our examples, showing that these examples are new.

Given an affine variety $X$, a morphism ${\it\phi}:X\rightarrow X$, a point ${\it\alpha}\in X$, and a Zariski-closed subset $V$ of $X$, we show that the forward ${\it\phi}$-orbit of ${\it\alpha}$ meets $V$ in at most finitely many infinite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the dynamical Mordell–Lang conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces.

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

Let ($M$, ${\rm\Omega}$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^{l}$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose that $N$ is controllable and that its stable and unstable bundles are trivial. We consider a $C^{1}$-submanifold ${\rm\Delta}$ of $M$ whose dimension is equal to the dimension of a fiber of the unstable bundle of $T_{N}M$. We suppose that ${\rm\Delta}$ transversely intersects the stable manifold of $N$. Then, we prove that for all ${\it\varepsilon}>0$, and for $n\in \mathbb{N}$ large enough, there exists $x_{n}\in N$ such that $f^{n}({\rm\Delta})$ is ${\it\varepsilon}$-close, in the $C^{1}$ topology, to the strongly unstable manifold of $x_{n}$. As an application of this ${\it\lambda}$-lemma, we prove the existence of shadowing orbits for a finite family of invariant minimal sets (for which we do not assume any regularity) contained in a normally hyperbolic manifold and having heteroclinic connections. As a particular case, we recover classical results on the existence of diffusion orbits (Arnold’s example).

In this article, we prove that for all pairs of primitive Pisot or uniform substitutions with the same dominating eigenvalue, there exists a finite set of block maps such that every block map between the corresponding subshifts is an element of this set, up to a shift.

This work focuses on the persistence of lower-dimensional tori with prescribed frequencies and singular normal matrices in reversible systems. By the Kolmogorov–Arnold–Moser theory and the special structure of unperturbed nonlinear terms in the differential equation, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of X. Wang et al [Degenerate lower dimensional tori in reversible systems. J. Math. Anal. Appl.387 (2012), 776–790].

In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the small parameter, we prove that the system is reducible for most of the sufficiently small parameters in the sense of the Lebesgue measure.