No-shadowing for singular hyperbolic sets with a singularity

Abstract

We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property. Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic.

Topological aspects of incompressible flows

“…geometric Lorenz flows or more generally singular-hyperbolic flows) which do not satisfy the shadowing property (see e.g. [40]). Indeed, while these flows are C 1 -persistent according to [25], they can be C 0 -approximated by structurally stable ones allowing to go ahead with Odani's strategy.…”

On multifractal analysis and large deviations of singular hyperbolic attractors

“…The latter relies ultimately on the uniqueness of equilibrium states for Hölder continuous potentials, a question which in such generality remains widely open. Moreover, singular hyperbolic attractors seem not to display any gluing orbit property, as hinted by [12,56,66]. We first show that the horseshoe approximation technique, valid for C r -generic geometric Lorenz attractors and C 1 -generic singular hyperbolic attractors, is enough to show that the level sets and irregular sets of such singular hyperbolic attractors inherit the properties of the corresponding objects for special classes of horseshoes approximating them.…”

“…Indeed, while most constructions of fractal sets with high entropy involve the use of some specification-like property, the presence of hyperbolic singularities constitutes an obstruction for specification (see e.g. [56,66]). In this direction, a standard argument is to establish the variational principle for saturated sets of generic points.…”

On multifractal analysis and large deviations of singular-hyperbolic attractors

“…The latter relies ultimately on the uniqueness of equilibrium states for Hölder continuous potentials, a question which in such generality remains widely open. Moreover, singular hyperbolic attractors seem not to display any gluing orbit property, as hinted by [11,47,54]. We first show that the horseshoe approximation technique, valid for C r -generic geometric Lorenz attractors and C 1 -generic singular hyperbolic attractors, is enough to show that the level sets and irregular sets of such singular hyperbolic attractors inherit the properties of the corresponding objects for special classes of horseshoes approximating them.…”

“…Indeed, while most constructions of fractal sets with high entropy involve the use of some specification-like property, the presence of hyperbolic singularities constitutes an obstruction for specification (see e.g. [47,54]). In this direction, a standard argument is to establish the variational principle for saturated sets of generic points.…”