Lipschitz contact equivalence of function germs in R^2

Abstract

In this paper we study Lipschitz contact equivalence of continuous function germs in the plane definable in a polynomially bounded o-minimal structure, such as semialgebraic and subanalytic functions. We partition the germ of the plane at the origin into zones where the function has explicit asymptotic behavior. Such a partition is called a pizza. We show that each function germ admits a minimal pizza, unique up to combinatorial equivalence. We show then that two definable continuous function germs are definably Lipschitz contact equivalent if and only if their corresponding minimal pizzas are equivalent.

Lipschitz geometry of pairs of normally embedded Hölder triangles

“…Proposition 2.12. (See [3].) Let T be a β-Hölder triangle which is a pizza slice associated with a non-negative Lipschitz function f , such that Q = Q f (T ) is not a point.…”

“…Definition 2.13. (See [3].) Let f be a non-negative Lipschitz function defined on an oriented β-Hölder triangle T .…”

“…For q ∈ Q f (Z) let µ Z,f (q) be the set of exponents µ Z (γ, f ), where γ ∈ Z is any arc such that ord γ f = q. It follows from [3] that, for each q ∈ Q f (Z), the set µ Z,f (q) is finite. This defines a multivalued width function µ Z,f : Q f (Z) → F ∪ {∞}.…”

“…The outer Lipschitz geometry of semialgebraic surface germs is considerably more complicated, and outer bi-Lipschitz classification of semialgebraic surface germs is still work in progress. An important step towards such classification was made in [3], where classification of the germs at the origin of R 2 of semialgebraic (or, more generally, definable in a polynomially bounded o-minimal structure) Lipschitz functions with respect to contact Lipschitz equivalence relation was suggested. It was based on a complete combinatorial invariant of contact Lipschitz equivalence, called pizza.…”

Lipschitz geometry and combinatorics of abnormal surface germs

“…Classification of surface germs with respect to the outer metric is much more complicated. The first step towards the outer metric classification, classification of semialgebraic functions with respect to K-Lipschitz equivalence, was made in [2]. It is equivalent to classification of relatively simple surface germs, each of them being the union of the real plane and a graph of a semialgebraic function defined on that plane.…”