Topological aspects of incompressible flows

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.