(SSP) geometry with directional homeomorphisms

Abstract

In a previous paper [6] we discussed several directional properties of sets satisfying the sequence selection property, denoted by (SSP) for short, and developed the (SSP) geometry via bi-Lipschitz transformations. In this paper we introduce the notion of directional homeomorphism and show that we can develop also the (SSP) geometry with directional transformations. For many important results proved in [6] for bi-Lipschitz homeomorphisms we describe the analogues for directional homeomorphisms as well.

Stabilisation of geometric directional bundle for a subanalytic set

“…We also give the definition of the geometric directional bundle and give some illustrative examples in §3. In our approach to Lipschitz geometry, using directional properties, we employ a basic property stated in Lemma 4.1 and Remark 4.2 (also discussed in [6]- [9]). In §4 we discuss a similar property for the geometric directional bundle, described by the Theorem 4.7.…”

Stabilisation of geometric directional bundle for a subanalytic set

“…In a series of papers [3,4,5,6,7], in order to introduce several local bi-Lipschitz invariants, we investigated several local directional properties. Subsequently, in order to introduce global bi-Lipschitz invariants, we have globalised the local directional properties in [8].…”

Notes on (SSP) sets

“…In this note, using Sampoio's idea, we generalise his main result in [9] and the aforementioned main result in [5] to the case of the (SSP ) setting. Although the proofs are in the spirit of [8] , basically the same as in [9] (at times even simpler), due to the wide potential applications, we believe that it is still worth mentioning this generalisation.…”

“…In proving that, we introduced and essentially used the notion of sequence selection property, denoted by (SSP ) for short. Subsequently we have published three more papers [6], [7] and [8], where we proved essential directional properties of sets satisfying (SSP ) with respect to bi-Lipschitz homeomorphisms. For instance we proved the transversality theorem in the singular case and two types of (SSP ) structure preserving theorems ( [7]), and we introduced the notion of directional homeomorphism, proving a unified (SSP ) structure preserving theorem with directional homeomorphisms ( [8]).…”

“…

In [8] we investigate the directional behaviour of bi-Lipschitz homeomorphisms h : (R n , 0) → (R n , 0) for which there exist the limits lim n→∞ nh( x n ), denoted by h(x). The existence of such h(x) makes trivial to see that h(D(A)) = D(h(A)) for arbitrary set-germs A at 0 ∈ R n .Recently, J. Edson Sampaio made the remarkable observation ([9]) that we always can assume the existence of a subsequence n i ∈ N, such that lim ni→∞ n i h( x ni ) = dh(x) (in his notation) and this dh, although not so strong as h, behaves as well directional-wise for subanalytic sets.

…”