We study limit sets of stable cellular automata from a symbolic dynamics point of view, where they are a special case of sofic shifts admitting a steady epimorphism. We prove that there exists a right-closing almost-everywhere steady factor map from one irreducible sofic shift onto another one if and only if there exists such a map from the domain onto the minimal right-resolving cover of the image. We define right-continuing almost-everywhere steady maps, and prove that there exists such a steady map between two sofic shifts if and only if there exists a factor map from the domain onto the minimal right-resolving cover of the image. To translate this into terms of cellular automata, a sofic shift can be the limit set of a stable cellular automaton with a right-closing almost-everywhere dynamics onto its limit set if and only if it is the factor of a full shift and there exists a right-closing almost-everywhere factor map from the sofic shift onto its minimal right-resolving cover. A sofic shift can be the limit set of a stable cellular automaton reaching its limit set with a right-continuing almost-everywhere factor map if and only if it is the factor of a full shift and there exists a factor map from the sofic shift onto its minimal right-resolving cover. Finally, as a consequence of the previous results, we provide a characterization of the almost of finite type shifts (AFT) in terms of a property of steady maps that have them as range.

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on ${\ell }^{p} ( \mathbb{Z} )$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper density. Secondly, we show that there exists an operator which is $ \mathcal{U} $-frequently hypercyclic but not frequently hypercyclic, and that there exists an operator which is frequently hypercyclic but not distributionally chaotic. These (surprising) counterexamples are given by weighted shifts on ${c}_{0} $. The construction of these shifts lies on the construction of sets of positive integers whose difference sets have very specific properties.

It is well known that optimal transport plans are cyclically monotone. The reverse implication that cyclically monotone transport plans are optimal needs some assumptions and the proof is non-trivial even if the costs are given by the squared euclidean distance on ${ \mathbb{R} }^{n} $. We establish this result as a corollary to the ergodic theorem.

A subset $A$ of $ \mathbb{N} $ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $\mathop{({x}_{n} )}\nolimits_{n\in \mathbb{N} } $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing rich combinatorial properties: each central set contains arbitrarily long arithmetic progressions and solutions to all partition regular systems of homogeneous linear equations. In this paper we investigate central sets in the framework of combinatorics on words. Using various families of uniformly recurrent words, including Sturmian words, the Thue–Morse word and fixed points of weak mixing substitutions, we generate an assortment of central sets which reflect the rich combinatorial structure of the underlying words. The results in this paper rely on interactions between different areas of mathematics, some of which have not previously been directly linked. They include the general theory of combinatorics on words, abstract numeration systems, and the beautiful theory, developed by Hindman, Strauss and others, linking IP-sets and central sets to the algebraic/topological properties of the Stone-Čech compactification of $ \mathbb{N} $.

For any $C^1$ diffeomorphism with dominated splitting, we consider a non-empty set of invariant measures that describes the asymptotic statistics of Lebesgue-almost all orbits. They are the limits of convergent subsequences of averages of the Dirac delta measures supported on those orbits. We prove that the metric entropy of each of these measures is bounded from below by the sum of the Lyapunov exponents on the dominating sub-bundle. As a consequence, if those exponents are non-negative, and if the exponents on the dominated sub-bundle are non-positive, those measures satisfy the Pesin entropy formula.

The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as coupled phase oscillators. In this paper, a bifurcation structure of the infinite-dimensional Kuramoto model is investigated. A purpose here is to prove the bifurcation diagram of the model conjectured by Kuramoto in 1984; if the coupling strength $K$ between oscillators, which is a parameter of the system, is smaller than some threshold ${K}_{c} $, the de-synchronous state (trivial steady state) is asymptotically stable, while if $K$ exceeds ${K}_{c} $, a non-trivial stable solution, which corresponds to the synchronization, bifurcates from the de-synchronous state. One of the difficulties in proving the conjecture is that a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, has the continuous spectrum on the imaginary axis. Hence, the standard spectral theory is not applicable to prove a bifurcation as well as the asymptotic stability of the steady state. In this paper, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid the continuous spectrum on the imaginary axis. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition consisting of a countable number of eigenfunctions on a space of generalized functions. The semigroup generated by the linear operator will be estimated with the aid of the spectral theory on a rigged Hilbert space to prove the linear stability of the steady state of the system. The center manifold theory is also developed on a space of generalized functions. It is proved that there exists a finite-dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimension because of the continuous spectrum on the imaginary axis. These results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto.

For an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

We define a random iterated function system (RIFS) to be a finite set of (deterministic) iterated function systems (IFSs) acting on the same metric space. For a given RIFS, there exists a continuum of random attractors corresponding to each sequence of deterministic IFSs. Much work has been done on computing the ‘almost sure’ dimensions of these random attractors. Here we compute the typical dimensions (in the sense of Baire) and observe that our results are in stark contrast to those obtained using the probabilistic approach. Furthermore, we examine the typical Hausdorff and packing measures of the random attractors and give examples to illustrate some of the strange phenomena that can occur. The only restriction we impose on the maps is that they are bi-Lipschitz and we obtain our dimension results without assuming any separation conditions.

Let $f$ be a transitive homeomorphism of the two-dimensional torus in the homotopy class of the identity. We show that a lift of $f$ to the universal covering is transitive if and only if the rotation set of the lift contains the origin in its interior.

We investigate a parametric extension of the classical $s$-dimensional Halton sequence where the bases are special Pisot numbers. In a one-dimensional setting the properties of such sequences have already been investigated by several authors. We use methods from ergodic theory in order to investigate the distribution behavior of multidimensional versions of such sequences. As a consequence it is shown that the Kakutani–Fibonacci transformation is uniquely ergodic.

We show that any non-zero orbit under a non-compact, simple, irreducible linear group is dense in the Bohr compactification of the ambient space.

We consider special flows over the rotation by an irrational $\alpha $ under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and sum of jumps not equal to zero. We show that all such flows are weakly mixing. Under the additional assumption that $\alpha $ has bounded partial quotients, we study the weak Ratner property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is a classical result that the flows under consideration are not mixing, one more condition on the set of jumps turns out to be sufficient to obtain the absence of partial rigidity, hence mild mixing of such flows.

We construct a curve in the unstable foliation of an Anosov diffeomorphism such that the holonomy along this curve is defined on all of the corresponding stable leaves.

We present a method, based on commutator methods, for the spectral analysis of uniquely ergodic dynamical systems. When applicable, it leads to the absolute continuity of the spectrum of the corresponding unitary operators. As an illustration, we consider time changes of horocycle flows, skew products over translations and Furstenberg transformations. For time changes of horocycle flows we obtain absolute continuity under assumptions weaker than those to be found in the literature, and for skew products over translations and Furstenberg transformations we obtain countable Lebesgue spectrum under assumptions not previously covered in the literature.

We obtain large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. In particular, a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some applications to the study of the convergence of Lyapunov exponents are given.

We give a natural notion of non-degeneracy for singular points of integrable non-Hamiltonian systems, and show that such non-degenerate singularities are locally geometrically linearizable and deformation rigid in the analytic case. We conjecture that the same result also holds in the smooth case, and prove this conjecture for systems of type $(n, 0)$, i.e. $n$ commuting smooth vector fields on an $n$-manifold.