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In this paper, we address the issue of synchronization of coupled systems, introducing concepts of local and global synchronization for a class of systems that extend the model of coupled map lattices. A criterion for local synchronization is given; numerical experiments are exhibited to illustrate the criteria and also to raise some questions in the end of the text.

This article is a follow up of our recent works [7, 8], and here we discuss the relation between the gluing orbit property and partial hyperbolicity. First we prove that a partially hyperbolic diffeomorphism with two saddles with different index, and such that the stable manifold of one of these saddles coincides with the strongly stable leaf does not satisfy the gluing orbit property. In particular, the examples of C 1 -robustly transitive diffeomorphisms introduced by Man˜e [ ´ 20] do not satisfy the gluing orbit property. We also construct some families of partially hyperbolic skew-products satisfying the gluing orbit property and derive some estimates on their quantitative recurrence., MJT was partially supported by FCT (Fundacao para a Ciencia e a Tecnologia-Portugal) within the Projects UIDB/00013/2020 and UIDP/00013/2020 and by the Project 'New trends in Lyapunov exponents' (PTDC/MAT-PUR/29126/2017). PV was partially supported by CNPq-Brazil, by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020, and by Fundacao para a Ciencia e a Tecnologia (FCT) - Portugal, through the grant CEECIND/03721/2017 of the Stimulus of Scientific Employment, Individual Support 2017 Call.

We say that a convex planar billiard table B is C^2-stably expansive on a fixed open subset U of the phase space if its billiard map f_B is expansive on the maximal invariant set Λ_(B,U) = ∩_(n∈Z ) f_B^n(U), and this property holds under C^2 perturbations of the billiard table. In this note we prove for such billiards that the closure of the set of periodic points of f_B in Λ_(B,U) is uniformly hyperbolic. In addition we show that this property also holds for a generic choice among billiards which are expansive., The authors were partially funded by the project "New Trends in Lyapunov Exponents" PTDC/MAT-PUR/29126/2017 financed by Fundacao para a Ciencia e a Tecnologia, Portugal.MB would also like to thank CMUP for providing the necessary conditions under which this work was developed. JLD was also partially funded by the Project CEMAPRE-UID/MULTI/00491/2019 financed by Fundacao para a Ciencia e a Tecnologia. MJT was also partially financed by national funds through Fundacao para a Ciencia e a Tecnologia within the projects UIDB/00013/2020 and UIDP/00013/2020.

In this article we approach some of the basic questions in topological dynamics, concerning periodic points, transitivity, the shadowing and the gluing orbit properties, in the context of C0-incompressible flows generated by Lipschitz vector fields. We prove that a C0-generic incompressible and fixed-point free flow satisfies the periodic shadowing property, it is transitive and has a dense set of periodic points in the non-wandering set. In particular, a C0-generic fixed-point free incompressible flow satisfies the reparametrized gluing orbit property. We also prove that C0-generic incompressible flows satisfy the general density theorem and the weak shadowing property, moreover these are transitive., The authors would like to thank the referee for the careful reading and comments. The authors were partially supported by the Project 'New trends in Lyapunov exponents' (PTDC/MATPUR/29126/2017) . MB was partially supported by FCTFundacAo para a Ciencia e a Tecnologia, through Centro de Matematica e Aplicacoes (CMAUBI) , Universidade da Beira Interior, project UID/00212/2020. MB would also like to thank CMUP for providing the necessary conditions in which this work was developed. MJT was partially financed by Portuguese Funds through FCT (FundacAo para a Ciencia e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020. PV was supported by CMUP (UID/MAT/00144/2013) , which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020, and by FundacAo para a Ciencia e Tecnologia (FCT) Portugal, through the grant CEECIND/03721/2017 of the Stimulus of Scientific Employment, Individual Support 2017 Call.

We prove that the closure of the closed orbits of a generic geodesic flow on a closed Riemannian n >= 2 dimensional manifold is a uniformly hyperbolic set if the shadowing property holds C2-robustly on the metric. We obtain analogous results using weak specification and the shadowing property allowing bounded time reparametrization., The authors were partially supported by the Project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/2017). MB was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. JLD was partially supported by the Project CEMAPRE - UID/MULTI/00491/2019 financed by FCT/MCTES through national funds. MJT was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the Fundação para a Ciência e a Tecnologia, through the Project UID/MAT/00013/2013.

We prove that chaotic flows (i.e. flows that satisfy the shadowing property and have a dense subset of periodic orbits) satisfy a reparametrized gluing orbit property similar to the one introduced in [7]. In particular, these are strongly transitive in balls of uniform radius. We also prove that the shadowing property for a flow and a generic time-t map, and having a dense subset of periodic orbits hold for a C0-Baire generic subset of Lipschitz vector fields, that generate continuous flows. Similar results also hold for C0-generic homeomorphisms and, in particular, we deduce that chain recurrent classes of C0-generic homeomorphisms have the gluing orbit property., The authors would like also to thank the reviewer for his/her careful reading of the manuscript and the helpful comments. MB was partially supported by FCT - ‘Funda¸c˜ao para a Ciˆencia e a Tecnologia’, through CMA-UBI, project UID/MAT/00212/2013. He would also like to thank CMUP for providing the necessary conditions in which this work was developed. MJT was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the ‘Funda¸c˜ao para a Ciˆencia e a Tecnologia’, through the Project UID/MAT/00013/2013. PV was partially supported by BREUDS and CNPq-Brazil., info:eu-repo/semantics/publishedVersion

We prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the specification properties. Despite the Hamiltonian nature of the geodesic flow, the arguments in the present paper differ completely from those used in Bessa et al. (2013) for Hamiltonian systems., The authors would like to thank the anonymous referee for several helpful comments. MB was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through CMA-UBI, project UID/MAT/00212/2013. JLD was partially supported by the Project CEMAPRE-UID/MULTI/00491/2013 financed by FCT/MEC through national funds. MJT was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the ‘Fundação para a Ciência e a Tecnologia’, through the Project UID/MAT/00013/2013., info:eu-repo/semantics/publishedVersion

The notions of shadowing, specification and gluing orbit property differ substantially for discrete and continuous time dynamical systems. In the present paper we continue the study of the topological and ergodic properties of continuous flows with the (reparametrized) periodic and nonperiodic gluing orbit properties initiated in [3]. We prove these flows satisfy a weak mixing condition with respect to balls and, if the flow is Komuro expansive, the topological entropy is a lower bound for the exponential growth rate of periodic orbits. Moreover, we show that periodic measures are dense in the set of all invariant probability measures and that ergodic measures are generic. Furthermore, we prove that irrational rotations and some minimal flows on tori and circle extensions over expanding maps satisfy gluing orbit properties, thus emphasizing the difference of this property with respect the notion of specification., Acknowledgements. MJT was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Fundação para a Ciência e a Tecnologia”, through the Project ˆ UID/MAT/00013/2013. PV was partially supported by CNPq-Brazil and BREUDS. The authors are grateful to M. Bessa and X. Tian for useful conversations and pointing some references, and to the anonymous referee for useful suggestions to shorten the manuscript., info:eu-repo/semantics/publishedVersion

In this paper we prove a Lusin theorem for the space of Sobolev-(1,p) volume preserving homeomorphism on closed and connected n-dimensional manifolds, n >= 3, for pn this result is not true., The authors would like to thank the anonymous referee for the careful reading of the manuscript and for giving very helpful comments and suggestions. AA and MJT were partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Fundação para a Ciência e a Tecnologia”, through the Project UID/MAT/ 00013/2013. MB was partially supported by FCT - ‘Fundação para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013.

Given a closed Riemannian manifold, we prove the C 0 -general density theorem for continuous geodesic flows. More precisely, we prove that there exists a residual (in the C 0 -sense) subset of the continuous geodesic flows such that, in that residual subset, the geodesic flow exhibits dense closed orbits.

Both authors would like to thank Armando P. Machado and Alessandro Margheri for valuable suggestions on the problems related to the Sturm-Liouville section. The first author was supported by "Fundacao para a Ciencia e a Tecnologia" through the Program POCI 2010 and the Project "Randomness in Deterministic Dynamical Systems and Applications" (PTDC-MAT-105448-2008). The second author was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the "Fundacao para a Ciencia e a Tecnologia", through the Project PEstOE/MAT/UI0013/2014.

For a class of semigroups of stochastic dynamical systems, $x\mapsto P_x$, where $x$ denotes a state and $P_x$ the state probability transition, we relate its spectral stability with the combinatorial stability of the underlying non-deterministic dynamics, associated to the point-set map $x\mapsto \text{ supp}(P_x)$.

A space of non-deterministic dynamical systems of Markov type on compact manifolds is considered. This is a natural space for stochastic perturbations of maps. For such systems, both the combinatorial stability, of the periodic attractors, and the spectral stability, of the invariant measures, are characterized and its genericity established.

L.A. Bunimovich and B.Z. Webb developed a theory for isospectral graph reduction. We make a simple observation regarding the relation between eigenvectors of the original graph and its reduction, that sheds new light on this theory. As an application we propose an updating algorithm for the maximal eigenvector of the Markov matrix associated to a large sparse dynamical network., The first author was partially supported by Fundacao para a Ciencia e a Tecnologia, PEst, OE/MAT/UI0209/2011.The second author was partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the "Fundacao para a Ciencia e a Tecnologia", through the Project PEstOE/MAT/UI0013/2014.

We prove that the boundary of an r-regular set is a codimension one manifold of class C1., MJT was partially financed by FEDER Funds through "Programa Operacional Factores de Competitividade - COMPETE'', Fundação para a Ciência e a Tecnologia (FCT) - PEst-C/MAT/UI0013/2011.

We prove that a Hamiltonian system H \in C^2(M,R) is globally hyperbolic if any of the following statements holds: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C^2-generic Hamiltonian H, the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits, forms a dense subset of M. As a consequence, any robustly transitive regular energy hypersurface of a C^2-Hamiltonian is partially hyperbolic. Finally, we prove that stably weakly shadowable regular energy hypersurfaces are partially hyperbolic., MJT was partially financed by FEDER Funds through "Programa Operacional Factores de Competitividade - COMPETE'' and by Portuguese Funds through FCT - "Fundação para a Ciência e a Tecnologia'', within the Project PEst-C/MAT/UI0013/2011.

We prove that a Hamiltonian star system, defined on a 2d-dimensional symplectic manifold M (d⩾2), is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians. This generalizes the 4-dimensional results in Bessa et al. (2010) [5]., FEDER Funds - "Programa Operacional Factores de Competitividade - COMPETE", Fundação para a Ciência e a Tecnologia (FCT) - Project PEst-C/MAT/UI0013/2011

This paper focuses on the concept of the seamless mobility (provided by vertical handover mechanisms) in heterogeneous Future Internet Networks in the context of the HURRICANE project. We discuss future open collaborative network architectures and propose a set of entities that ensure seamless mobility and service continuity, from infrastructural/operational and business standpoints: an 802.21-like implementation of vertical handover, supported by an innovative business modelling software package, which replaces the traditional operations support stack with a more dynamic, marketplace-oriented system. We also discuss, an approach to distributed, policy-based, vertical handover management and a model for such policies, based on the monitorization and analysis of Call Detail Records that reveals users’ behavioural patterns and allows personalization., Project INFSO-ICT-216006 HURRICANE (Handovers for Ubiquitous and optimal bRoadbandconnectIvity among CooperAtive Networking Environments), Fundação para a Ciência e a Tecnologia (FCT) - Pluriannual Funding Program

We introduce and study, from a combinatorial-topological viewpoint, some semigroups of continuous non-deterministic dynamical systems. Combinatorial stability, i.e. the persistence of the combinatorics of the attractors, is characterized and its genericity established. Some implications on topological (deterministic) dynamics are drawn.

We prove that given a non-wandering point of a Sobolev-(1,p) homeomorphism we can create closed trajectories by making arbitrarily small perturbations. As na application, in the planar case, we obtain that generically the closed trajectories are dense in the non- wandering set., The first and fourth authors were partially supported by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the "Fundacao para a Ciencia e a Tecnologia", through the Project UID/MAT/00013/2013. The third and fourth authors were partially supported by the Project "New trends in Lyapunov exponents" (PTDC/MAT-PUR/29126/2017).

We extend the theory of isospectral reductions of L. Bunimovich and B. Webb to infinite graphs, and describe an application of this extension to the problems of existence and approximation of stationary measures on infinite graphs., The authors would like to thank the referee for the very useful comments and suggestions that significantly improved this paper. AB would like to thank the hospitality of Universidade de Lisboa. PD was partially supported by Portuguese Funds through FCT (Fundacao para a Ciencia e a Tecnologia) within the Projects UIDB/04561/2020 and UIDP/04561/2020. MJT was partially financed by Portuguese Funds through FCT (Fundacao para a Ciencia e a Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020. AB, PD and MJT were partially supported by the Project "New trends in Lyapunov exponents"(PTDC/MATPUR/29126/2017).