A shadowable point for a flow is a point where the shadowing lemma holds for pseudo-orbits passing through it. We prove that this concept satisfies the following properties: the set of shadowable points is invariant and a Gδ set. A flow has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is shadowable. The chain recurrent points which are shadowable are exactly those that can be are approximated by periodic points when the flow is expansive. These results extends those presented in Morales (Dyn Syst. 2016;31(3):347–356). We study the relations between shadowable points of a homeomorphism and the shadowable points of its suspension flow. We characterize the set of forward shadowable points for transitive flows and chain transitive flows. We prove that the geometric Lorenz att...

We use the pointwise Lipschitz constant to define an upper Lyapunov exponent for maps on metric spaces different to that given by Kifer ['Characteristic exponents of dynamical systems in metric spaces', Ergodic Theory Dynam. Systems 3(1) (1983), 119-127]. We prove that this exponent reduces to that of Bessa and Silva on Riemannian manifolds and is not larger than that of Kifer at stable points. We also prove that it is invariant along orbits in the case of (topological) diffeomorphisms and under topological conjugacy. Moreover, the periodic orbits where this exponent is negative are asymptotically stable. Finally, we estimate this exponent for certain hyperbolic homeomorphisms.

RM was partially supported by Fondecyt-Concytec contract 100-2018 , HV was partially supported by Universidad Nacional de Ingeniería FC-PF-33-2021 and P-CC-2021-000666 . CAM was partially supported by CNPq -Brazil and the NRF Brain Pool Grant funded by the Korea government (No. 2020H1D3A2A01085417 ).

KL was partially supported by NRF grant No. 2018R1A2B3001457 . CAM was partially supported by CNPq -Brazil No. 307776/2019-0 and the NRF Brain Pool Grant No. 2020H1D3A2A01085417 . HV was partially supported by Universidad Nacional de Ingeniería P-CC-2021-000666 , FC-PF-33-2021 and Fondecyt-Concytec contract 100-2018 .

Kato [5] and Artigue [3] merged the theory of expansive systems [10] and foliations with the continuum theory [14]. Here we merge the expansive systems but with the descriptive set theory [6] instead. More precisely, we define meagre-expansivity for both homeomorphisms and measures by requiring the interior of the dynamical balls up to some prefixed radio to be either empty or with zero measure respectively. We first prove that every cw-expansive homeomorphism of a locally connected metric space without isolated points is meagre-expansive (but not conversely). Second that a homeomorphism of a metric space is meagre-expansive if and only if every Borel probability measure is meagre-expansive. Next that the space of meagre-expansive measures of a homeomorphism of a compact metric space X is an Fσ subset of the space of Borel probability measures equipped with the weak* topology. In the se...

We discuss the stability and the expansivity of the tent map f: [0, 1] ? [0, 1] defined by f(x) = 2 min{x, 1 ? x} for 0 ? x ? 1. Indeed, we show that f is neither topologically stable nor orbit shift topologically stable nor countably-expansive but is cw-topologically stable, orbit shift cw-expansive, and orbit shift ?-persistent. © 2020 American Mathematical Society

We extend the classical Banach-Mazur distance [3] from Banach spaces to linear operators between these spaces. We prove in the finite dimensional case that the corresponding topology is metrizable, complete, separable and locally compact. Furthermore, we prove that the Banach-Mazur compactum embeds isometrically into the resulting topological space. (C) 2020 Elsevier B.V. All rights reserved.

We prove that every finite-expansive homeomorphism with the shadowing property has a kind of stability. This stability will be good enough to imply both the shadowing property and the denseness of periodic points in the chain recurrent set. Next we analyze the N-shadowing property which is really stronger than the multishadowing property in Cherkashin and Kryzhevich (Topol Methods Nonlinear Anal 50(1): 125–150, 2017). We show that an equicontinuous homeomorphism has the N-shadowing property for some positive integer N if and only if it has the shadowing property. © 2021, Sociedade Brasileira de Matemática.

The authors would like to thank Carlos A. Morales, for useful talks on the subject of expansiveness and shadowing. Also, the authors would like to thank Professor H. Miranda and the anonymous reviewers for their valuable comments that helped to improve the final version of the article. The first author was partially supported by CONICYT PFCHA/DOCTORADO NACIONAL/2017-21170110 and Agencia Nacional de Investigaci?n y Desarrollo-ANID, Chile, project FONDECYT 1181061. The second author was partially supported by Agencia Nacional de Investigaci?n y Desarrollo-ANID, Chile, project FONDECYT 1181061, by Universidad del B?o-B?o, Chile, project 196108 GI/C, and by Programa do Postdoutorado Ver?o 2017-2019, IMPA, Rio de Janeiro, Brasil. The last author was partially supported by FONDECYT (Per?) contract 100?2018.