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We obtain a lower bound for the entropy of a Borel probability measure (not necessarily invariant) with respect to an upper semicontinuous set-valued map as the product of the lower dimension of the measure and the logarithmic expansion rate. This is a generalization of the well-known measure-preserving single-valued case. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We introduce the definitions of expansivity and topological stability for homeomorphisms on fuzzy metric spaces. We show some basic properties of fuzzy expansive homeomorphisms. Moreover we prove Walters's theorem in the context of fuzzy metric spaces, i.e., a fuzzy expansive system with the fuzzy shadowing property is fuzzy topologically stable. [ABSTRACT FROM AUTHOR]

We discuss the stability and the expansivity of the tent map ƒ : [0,1] → [0,1] defined by ƒ(x) = 2min {x,1−x} for 0 ≤ x ≤ 1. Indeed, we show that ƒ is neither topologically stable nor orbit shift topologically stable nor countably-expansive but is cw-topologically stable, orbit shift cw-expansive , and orbit shift α-persistent. [ABSTRACT FROM AUTHOR]

The geometric Lorenz attractor is an attractor set constructed in such a way that it satisfies the main qualitative properties evidenced on the Lorenz system equations, particularly the fact that this attractor is a robustly transitive set. In this paper we prove the C 1 -robust transitivity by using geometric properties for singular hyperbolic sets and without the assumption of the uniformly linearizing coordinates around the singularity. [ABSTRACT FROM AUTHOR]

We study the polynomial entropy of homeomorphisms on compact metric spaces. We construct a homeomorphism on a compact metric space with vanishing polynomial entropy that it is not equicontinuous. Also we give examples with arbitrarily small polynomial entropy. Finally, we show that expansive homeomorphisms and positively expansive maps of compact metric spaces with infinitely many points have polynomial entropy greater than or equal to 1. [ABSTRACT FROM AUTHOR]

Abstract: We extend the concept of expansive measure [2] from homeomorphism to flows. We prove for continuous flows on compact spaces that every expansive measure has no singularities in the support, is aperiodic, is expansive with respect to time-T maps (but not conversely), remains expansive under topological equivalence, vanishes along the orbits and is natural under suspensions. We apply these properties to prove that there are no expansive flows (in the sense of [26]) of any closed surface. [Copyright &y& Elsevier]

Let M be a 3-manifold with boundary ∂M. Let X be a C∞, vector field on M, tangent to ∂M, exhibiting a singular cycle associated to a hyperbolic equilibrium σ∈∂M with real eigenvalues λss < λs < 0 < λu satisfying λs - λss - 2λu > 0. We prove under generic conditions and k large enough the existence of a Ck robust transitive set of X, that is, any Ck vector field Ck close to X exhibits a transitive set containing the cycle. In particular, C∞ vector fields exhibiting Ck robust transitive sets, for k large enough, which are not singular-hyperbolic do exist on any compact 3-manifold with boundary. [ABSTRACT FROM AUTHOR]