Marina Murillo-Arcila

Strong mixing measures for linear operators and frequent hypercyclicity

Abstract We construct strongly mixing invariant measures with full support for operators on F -spaces which satisfy the Frequent Hypercyclicity Criterion. For unilateral backward shifts on sequence spaces, a slight modification shows that one can even obtain exact invariant measures.

Mixing properties for nonautonomous linear dynamics and invariant sets

Abstract We study mixing properties (topological mixing and weak mixing of arbitrary order) for nonautonomous linear dynamical systems that are induced by the corresponding dynamics on certain invariant sets. The kinds of nonautonomous systems considered here can be defined using a sequence ( T i ) i ∈ N of linear operators T i : X → X on a topological vector space X such that there is an invariant set Y for which the dynamics restricted to Y satisfies a certain mixing property. We then obtain the corresponding mixing property on the closed linear span of Y . We also prove that the class of nonautonomous linear dynamical systems that are weakly mixing of order n contains strictly the corresponding class with the weak mixing property of order n + 1 .

Sharp values for the constants in the polynomial Bohnenblust-Hille inequality

In this paper we prove that the complex polynomial Bohnenblust–Hille constant for 2-homogeneous polynomials in is exactly . We also give the exact value of the real polynomial Bohnenblust–Hille constant for 2-homogeneous polynomials in . Finally, we provide lower estimates for the real polynomial Bohnenblust–Hille constant for polynomials in of higher degrees.

Dynamics of multivalued linear operators

Abstract We introduce several notions of linear dynamics for multivalued linear operators (MLO’s) between separable Fréchet spaces, such as hypercyclicity, topological transitivity, topologically mixing property, and Devaney chaos. We also consider the case of disjointness, in which any of these properties are simultaneously satisfied by several operators. We revisit some sufficient well-known computable criteria for determining those properties. The analysis of the dynamics of extensions of linear operators to MLO’s is also considered.

Linear dynamics of semigroups generated by differential operators

Abstract During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C0-semigroups of linear and continuous operators. We will review some of these notions and we will discuss basic properties of the dynamics of C0-semigroups. We will also study in detail the dynamics of the translation C0-semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C0-semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models.

Cantor Sets, Bernoulli Shifts and Linear Dynamics

Our purpose is to review some recent results on the interplay between the symbolic dynamics on Cantor sets and linear dynamics. More precisely, we will give some methods that allow the existence of strong mixing measures invariant for certain operators on Frechet spaces, which are based on Bernoulli shifts on Cantor spaces. Also, concerning topological dynamics, we will show some consequences for the specification properties.

Lebesgue regularity for differential difference equations with fractional damping

Let X be a Banach space. We provide necessary and sufficient conditions for existence and uniqueness of solutions belonging to the vector-valued space of sequences lp(Z, X) for the linearized part of equations that can be modeled in the form ∆u(n) + μ∆u(n) = Au(n) +G(u)(n) + f(n), n ∈ Z, α, β > 0, μ ≥ 0, where f ∈ lp(Z, X), A is a closed linear operator with domain D(A) defined on X and G is a nonlinear term. The operator ∆ denotes the fractional difference operator of order γ > 0 in the sense of Grünwald-Letnikov. Our class of models includes the discrete time Klein-Gordon, telegraph and Basset equations, among others differential difference equations of interest. We prove a simple criteria that show existence of solutions assuming that f is a small and that G is at less of quadratic order.

Dynamics on Binary Relations over Topological Spaces

The existence of chaos and the quest of dense orbits have been recently considered for dynamical systems given by multivalued linear operators. We consider the notions of topological transitivity, topologically mixing property, hypercyclicity, periodic points, and Devaney chaos in the general case of binary relations on topological spaces, and we analyze how they can be particularized when they are represented with graphs and digraphs. The relations of these notions with different types of connectivity and with the existence of Hamiltonian paths are also exposed. Special attention is given to the study of dynamics over tournaments. Finally, we also show how disjointness can be introduced in this setting.

An Analytical Model Based on Population Processes to Characterize Data Dissemination in 5G Opportunistic Networks

The scarcity of bandwidth due to the explosive growth of mobile devices in 5G makes the offloading messaging workload to Wi-Fi devices that use opportunistic connections, a very promising solution. Communications in mobile opportunistic networks take place upon the establishment of ephemeral contacts among mobile nodes using direct communication. In this paper, we propose an analytical model based on population processes to evaluate data dissemination considering several parameters, such as user density, contact rate, and the number of fixed nodes. From this model, we obtain closed-form expressions for determining the diffusion time, the network coverage, and the waiting time. Newer 5G wireless technologies, such as WiGig, can offer multi-gigabit speeds, low latency, and security-protected connectivity between nearby devices. We therefore focus our work on the impact of high-speed and short-range wireless communications technologies for data dissemination in mobile opportunistic networks. Moreover, we test whether the coexistence with a fixed infrastructure can improve content dissemination, and thus justify its additional cost. Our results show that, when user density is high, the diffusion is mainly performed through the opportunistic contacts between mobile nodes, and that the diffusion coverage is close to 100%. Moreover, the diffusion is fast enough to dynamically update the information among all the participating members, so users do not need to get closer to fixed spots for receiving updated information.

Lebesgue regularity for nonlocal time-discrete equations with delays

Abstract In this work we provide a new and effective characterization for the existence and uniqueness of solutions for nonlocal time-discrete equations with delays, in the setting of vector-valued Lebesgue spaces of sequences. This characterization is given solely in terms of the R-boundedness of the data of the problem, and in the context of the class of UMD Banach spaces.