LIPSCHITZ CELL DECOMPOSITION

IN O-MINIMAL STRUCTURES. I

Wiesław Pawłucki

Uniwersytet Jagielloński, Instytut Matematyki ul. Reymonta 4, 30-059 Kraków, Poland

September 4, 2007

Abstract. A main tool in studying topological properties of sets definable in ominimal structures is the Cell Decomposition Theorem. The present paper proposes its metric counterpart based on the idea of a Lipschitz cell. In contrast to earlier results, we give an algorithm of a Lipschitz cell decomposition involving only permutations of variables as changes of coordinates.

1.Introduction.

A main tool in studying topological properties of sets definable in o-minimal structures is the Cell Decomposition Theorem (cf. [vdD]). The present paper proposes its metric counterpart based on the idea of a Lipschitz cell, called here an M-cell. Of course, in general a decomposition into such cells requires linear changes of coordinate systems (cf. [K, P]). We will give an algorithm showing that in fact permutations of coordinates suffice as changes of coordinate systems.

The present article deals only with Lipschitz cell decomposition of open sets. The case of general o-minimal sets, easily reducible to the previous one, with some additional properties and applications will be treated in a separate paper.

Fix any o-minimal structure on a real closed field R (for the definition and fundamental properties of o-minimal structures the reader is referred to [vdD]). Let n be a positive integer.

A subset S of R^{n} will be called an (open) cell in R^{n} iff

(1.1)  S = {(x′, x_{n}) ∈ R^{n} : x′ ∈ Δ, φ_{1}(x′) < x_{n} < φ_{2}(x′)},

where x′ = (x_{1}, …, x_{n−1}), Δ is an open definable subset of R^{n−1}, every φ_{i} (i ∈ {1, 2}) is either a definable continuous function φ_{i} : Δ ⟶ R or φ_{i} ≡ −∞ or φ_{i} ≡ +∞ and, for each x′ ∈ Δ, φ_{1}(x′) < φ_{2}(x′).

2000 Mathematics Subject Classification: Primary 32B20, 14P10. Secondary 32S60, 51N20, 51F99.

Key words and phrases: M-cell, semi-M-cell, M-disc, Lipschitz mapping, cell decomposition.

Research partially supported by the grant of the Polish Ministry of Research and Education 0325/PO3/2004/27 and the European Community IHP-Network RAAG (HPNR-CT-2001-00271)

For any positive M ∈ R, a definable continuous function φ : Δ ⟶ R defined on an open subset Δ of R^{n−1} will be called an M-function iff

(1.2)  |\frac{∂φ}{∂x_{j}} (a)| ≤ M  (j ∈ \{1, …, n − 1\}),

at each point a ∈ Δ in a neighborhood of which φ is of class C^{1}.

An cell S in R^{n} will be called an M-cell (a semi-M-cell) iff, for each i ∈ {1, 2} (for at least one i ∈ {1, 2}), if φ_{i} is finite, it is an M-function. A cell S in R^{n} will be called a regular M-cell iff it is any open interval in the case n = 1 and, in the case n > 1, for each i ∈ {1, 2}, if φ_{i} is finite it is an M-function of class C^{1} on Δ and the projection Δ of S into R^{n−1} is a regular M-cell in R^{n−1}.

An M-cell will be called an M-disc iff it is any open interval in the case n = 1 and, in the case n > 1, the both φ_{i} (i ∈ {1, 2}) are finite and admit continuous extensions

(1.3)  φ_{i} : \overline{Δ} ⟶ R

onto the closure of Δ in R^{n−1}, and

(1.4)  φ_{1} = φ_{2} on ∂Δ.

Proposition 1. Let S be a regular M-cell in R^{n} and let φ : S ⟶ R be an Lfunction ( L > 0 ) of class C^{1}.

Then

(1) for any two different points a, b ∈ S, there is a definable continuous mapping

λ = (λ_{1}, …, λ_{n}) : [0 , |a − b|] ⟶ S

such that λ(0) = a, λ(|a − b|) = b and |λ′_{j}(t)| ≤ (j − 1)!M^{j−1}, for any j ∈ {1, …, n} and any t such that λ′_{j}(t) exists^{1};

(2) for any two points a, b ∈ S, |φ(a) − φ(b)| ≤ n!M^{n−1}L|a − b|.

Proof. (1) Let S be as in (1.1). Arguing by induction and assuming that a′ ≠ b′, one can find a mapping

ω = (ω_{1}, …, ω_{n−1}) : [0 , |a′ − b′|] ⟶ Δ

such that ω(0) = a′, ω(|a′ − b′|) = b′ and |ω′_{j}(τ)| ≤ (j − 1)!M^{j−1}, for any j ∈ {1, …, n − 1} and any τ such that ω′_{j}(τ) exists. Let ε > 0 be such that

φ_{1}(ω(τ)) + ε < φ_{2}(ω(τ)) − ε, for any τ ∈ [0 , |a′ − b′|],

and

φ_{1}(a′) + ε < a_{n} < φ_{2}(a′) − ε and φ_{1}(b′) + ε < b_{n} < φ_{2}(b′) − ε.

Now, it suffices to put

λ_{j}(t) = ω_{j}(t\frac{|a′ − b′|}{|a − b|}), for j ∈ \{1, …, n − 1\},

and

λ_{n}(t) = max\{φ_{1}(ω(t\frac{|a′ − b′|}{|a − b|})) + ε, min\{φ_{2}(ω(t\frac{|a′ − b′|}{|a − b|})) − ε, a_{n} + t\frac{b_{n} − a_{n}}{|a − b|} \}\}.

(2) follows from (1), by the Mean Value Theorem (see [vdD, Chapter 7, (2.3)]).

1|a − b| = \sqrt{∑_{j=1}^{n}(a_{j} − b_{j})^{2}}

Kurdyka-Parusiński Theorem ([K, P]). Any open definable subset G of R^{n} has a finite decomposition

G = S_{1} ∪ ⋯ ∪ S_{k} ∪ Σ,

where every S_{ν} is a regular M_{n}-cell in some linear coordinate system in R^{n} and Σ is nowhere dense, M_{n} being a constant depending only on n.

The aim of the present article is to give an algorithm showing that in fact permutations of coordinates are sufficient in the above theorem. We will prove simultaneously by induction on n the following three theorems.

Theorem 1_{n} (2_{n}, 3_{n}). Any open definable subset G of R^{n} has a finite decomposition

(1.5)  G = S_{1} ∪ ⋯ ∪ S_{k} ∪ Σ,

where every S_{ν} is an M_{1n}-cell (M_{2n}-disc, a regular M_{3n}-cell) in R^{n} after a permutation of coordinates and Σ is nowhere dense, M_{1n} (M_{2n}, M_{3n}) being a constant ≥ 1 depending only on n.

For simplicity we will often skip the adjective definable, when considering subsets of spaces R^{n} and mappings between such subsets. Also, we adopt the following conventions. A local property (w) of a mapping f : A ⟶ R^{m}, where A ⊂ R^{n}, is said to be satisfied almost everywhere iff there is a closed subset E of A such that dim E < dim A and (w) is satisfied at each point of A \ E. A finite sequence B_{1}, …, B_{k} of subsets of a set A ⊂ R^{n} is said to be an almost decomposition of A iff B_{ν} (ν = 1, …, k) are pairwise disjoint and dim (A \ (B_{1} ∪ ⋯ ∪ B_{k})) < dim A. This will be denoted by writing

A ≃ B_{1} ∪ ⋯ ∪ B_{k}.

Since Theorem 2_{n} together with 3_{n−1} easily imply both Theorems 1_{n} and 3_{n}, it suffices to derive first Theorem 1_{n} from Theorem 2_{n−1} and then Theorem 2_{n} from Theorems 1_{n}, 2_{n−1} and 3_{n−1}. From now on, we will assume that n ≥ 2 is fixed.

2. A preparation.

Lemma 1. If G ⊂ R^{n−1} is open and E ⊂ ∂G is closed of dimension < n − 2 and Theorem 2_{n−1} is true, then G has an almost decomposition

G ≃ Δ_{1} ∪ ⋯ ∪ Δ_{p},

where every Δ_{ν}, after a permutation of coordinates in R^{n−1}, is an M_{2n−1}-disc:

Δ_{ν} = {(x′′, x_{n−1}) : x′′ ∈ Ω_{ν}, σ_{ν}(x′′) < x_{n−1} < ρ_{ν}(x′′)}^{2},

such that the both (graphs of )^{3} σ_{ν} and ρ_{ν} are disjoint with E.

2x′′ = (x_{1}, …, x_{n−2})

3We will identify functions with their graphs.

4 WIESŁAW PAWŁUCKI

Proof. Take the projections

π_{j} : R^{n−1} ∋ (x_{1}, …, x_{n−1}) ↦ (x_{1}, …, x_{j−1}, x_{j+1}, …, x_{n−1}) ∈ R^{n−2},

for j ∈ {1, …, n − 1}, and set

Z = the closure of ⋃_{j} π_{j}^{−1}(π_{j}(E)).

Then dim Z ≤ n − 2 and it suffices to use Theorem 2_{n−1} to G \ Z.

As a corollary one easily gets (see [vdD]) the following Lemma 2. If G ⊂ R^{n−1} is open and φ : G ⟶ R is continuous, then G has an almost decomposition

G ≃ Δ_{1} ∪ ⋯ ∪ Δ_{p} ,

where every Δ_{ν}, after a permutation of coordinates in R^{n−1}, is an M_{2n−1}-disc

Δ_{ν} = {(x′′, x_{n−1}) : x′′ ∈ Ω_{ν}, σ_{ν}(x′′) < x_{n−1} < ρ_{ν}(x′′)}

such that φ|Δ_{ν} has a continuous extension

φ_{ν} : Δ_{ν} ∪ σ_{ν} ∪ ρ_{ν} ⟶ \overline{R} = R ∪ \{−∞, +∞\}

such that φ_{ν}(σ_{ν}) ⊂ R or φ_{ν}(σ_{ν}) = {−∞}, or φ_{ν}(σ_{ν}) = {+∞} and the same for ρ_{ν}.

Proposition 2. Let f : S ⟶ R be a definable C^{1}-function defined on a cell

S = {(x′, x_{n}) ∈ R^{n} : x′ ∈ Δ, φ(x′) < x_{n} < ψ(x′)}

in R^{n} such that φ : Δ ⟶ R is of class C^{1}.

Assume that \frac{∂f}{∂x_{n}} has a finite limit value^{4} at (almost) each point of φ ( for example, when \frac{∂f}{∂x_{n}} is bounded ).

Then there is a closed nowhere dense subset Z of φ such that f extends to a C^{1}-function

f : S ∪ (φ \ Z) ⟶ R

to S ∪ (φ \ Z) as a C^{1}-submanifold with boundary. Proof. It is left to the reader as an exercise (cf [vdD]).

4An element α ∈ \overline{R} is a limit value of a function g : S ⟶ R at a ∈ \overline{S} iff there is an arc γ : (0, 1) ⟶ S such that lim_{t→0} γ(t) = a and lim_{t→0} g(γ(t)) = α.

Lemma 3. Let L, M, N, P ∈ R be positive and let

G = {(x′, x_{n}) : x′ ∈ Δ, φ_{1}(x′) < x_{n} < φ_{2}(x′)}

be a semi-M-cell in R^{n} such that Δ is an N-cell in R^{n−1}, φ_{i} : Δ ⟶ R, for each i ∈ {1, 2},and the following conditions are satisfied almost everywhere in Δ:

(2.1)  |\frac{∂φ_{1}}{∂x_{j}} | ≤ M, for each j ∈ \{1, …, n − 1\};

(2.2)  |\frac{∂φ_{1}}{∂x_{n−1}} | < L < |\frac{∂φ_{2}}{∂x_{n−1}} |;

(2.3)  \begin{matrix}| & \frac{∂φ_{2}}{∂x_{j}} & | \\ | & \frac{∂φ_{2}}{∂x_{n−1}} & | \end{matrix} ≤ P, for each j ∈ \{1, …, n − 1\};

(2.4)  sgn \frac{∂φ_{2}}{∂x_{n−1}} = const.

Then G admits an almost decomposition

G ≃ S_{1} ∪ ⋯ ∪ S_{k},

where every S_{ν} is an M̃-cell, possibly after transposition (x_{n−1}, x_{n}), where M̃ is a positive constant depending only on L, M, N and P.

Proof. Put

Δ = {(x′′, x_{n−1}) : x′′ ∈ Ω, σ(x′′) < x_{n−1} < ρ(x′′)}.

One can assume that

(2.5)  \frac{∂φ_{2}}{∂x_{n−1}} > 0 ;

the other case will follow by a modification. Because of (2.2) and (2.5), it is clear that σ : Ω ⟶ R. By a subdivision of Ω one can assume that σ is of class C^{1} and that (2.2) is satisfied almost everywhere on every segment {(x′′, x_{n−1}) : σ(x′′) < x_{n−1} < ρ(x′′)}, where x′′ ∈ Ω and that φ_{i} admit continuous extensions

φ_{i} : Δ ∪ σ ⟶ R (i = 1, 2)

and

φ_{2} : Δ ∪ ρ ⟶ R ∪ {+∞}

such that φ_{2}(ρ) ⊂ R or φ_{2}(ρ) = {+∞}.

6 WIESŁAW PAWŁUCKI

By Proposition 2, φ_{1} is of class C^{1} almost everywhere on σ. Put

ψ(x′′, x_{n−1}) = φ_{1}(x′′, σ(x′′)) + L(x_{n−1} − σ(x′′)), for (x′′, x_{n−1}) ∈ Δ. Then ψ is an max(M + MN + LN, L)-function and φ_{1} < ψ < φ_{2}.

Now G ≃ S_{1} ∪ S_{2}, where S_{1} = {(x′, x_{n}) : φ_{1}(x′) < x_{n} < ψ(x′)} and S_{2} = {(x′′, x_{n−1}, x_{n}) : x′′ ∈ Ω, Φ_{1}(x′′, x_{n}) < x_{n−1} < Φ_{2}(x′′, x_{n})}, where

Φ_{2}(x′′, x_{n}) = \{ \begin{matrix}ψ^{−1}(x′′, x_{n}) = L^{−1}(x_{n} − φ_{1}(x′′, σ(x′′))) + σ(x′′), \\ if φ_{1}(x′′, σ(x′′)) < x_{n} < ψ(x′′, ρ(x′′)) \\ ρ(x′′), if ψ(x′′, ρ(x′′)) ≤ x_{n} < φ_{2}(x′′, ρ(x′′)) \end{matrix}

and

Φ_{1}(x′′, x_{n}) = \{ \begin{matrix}σ(x′′), & if φ_{1}(x′′, σ(x′′)) < x_{n} ≤ φ_{2}(x′′, σ(x′′)) \\ φ_{2}^{−1}(x′′, x_{n}), & if φ_{2}(x′′, σ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′)), \end{matrix}

where ψ^{−1} and φ_{2}^{−1} stand for inversions with respect to x_{n−1}. Lemma 4. Let A ⊂ R^{n−1} be open and let M ∈ R, M > 0. Let f_{α} : A ⟶ R (α ∈ {1, …, k + l}) be M-functions on A each of which has a continuous extension to \overline{A}:

f_{α} : \overline{A} ⟶ R.

Assume that for each a ∈ ∂A there are α ≤ k and β > k such that f_{β}(a) ≤ f_{α}(a).

Then the set

S = {(x′, x_{n}) ∈ A × R : max_{1≤α≤k} f_{α}(x′) < x_{n} < min_{k<β≤k+l} f_{β}(x′)}

is an M-disc in R^{n}. Proof. Indeed,

S = {(x′, x_{n}) ∈ B × R : max_{1≤α≤k} f_{α}(x′) < x_{n} < min_{k<β≤k+l} f_{β}(x′)},

where B is the natural projection of S to A. It is clear that max_{1≤α≤k} f_{α} = min_{k<β≤k+l} f_{β} on ∂B and the lemma follows. Lemma 5. Let α_{1}, α_{2} ∈ \overline{R}, α_{1} < α_{2} and let f, g, h : (α_{1}, α_{2}) ⟶ R be three continuous definable functions such that

(2.6)  g ≤ f on (α_{1}, α_{2});
(2.7)  for each i ∈ {1, 2}, if α_{i} ∈ R, then lim_{t→α_{i}} g(t) = lim_{t→α_{i}} h(t) ∈ R;