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On a small cube, a $C^{1}$ diffeomorphism distorts Jordan content by factors arbitrarily close to its linearized absolute determinant

Statement

Let $n \geq 1$ , let $g : U \to R^{n}$ be $C^{1}$ on an open set, let $a \in U$ , and suppose $A = D g (a)$ is invertible. For every $0 < ε < 1$ there is a closed cube $Q$ centred at $a$ , of positive radius and contained in $U$ , such that every Jordan set $E \subseteq Q$ has Jordan image and $∣ \det D g (a) ∣ (1 - ε)^{n} cont (E) \leq cont (g (E)) \leq ∣ \det D g (a) ∣ (1 + ε)^{n} cont (E) .$ The cube may be chosen inside any prescribed neighborhood of $a$ .

Facts & Assumptions

Given: The $C^{1}$ map, the point $a$ , invertible $A = D g (a)$ , and $0 < ε < 1$ .

[L1]

A linear endomorphism maps Jordan sets to Jordan sets and scales content by its absolute determinant (A linear endomorphism of $R^{n}$ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant).

[L2]

A map whose derivative is uniformly close enough to the identity sandwiches each sufficiently small cube between concentric contracted and expanded cubes (A $C^{1}$ map uniformly close to the identity derivative sandwiches a cube between contracted and expanded cubes).

[L3]

Jordan inner and outer content approximate Jordan sets by finite rectangular figures (Jordan inner and outer content and Jordan measurable bounded sets in $R^{m}$ ).

[L4]

A derivative bound on a convex open set gives a Lipschitz bound (On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ ); an everywhere-invertible derivative gives local $C^{1}$ inverses (The Euclidean inverse function theorem); Lipschitz self-maps preserve null sets (A Lipschitz map $R^{m} \to R^{m}$ sends null sets to null sets); and bounded sets are Jordan exactly when their boundaries are null (A bounded set in $R^{m}$ is Jordan measurable iff its boundary is null, equivalently of content zero).

[L5]

Jordan content is finitely additive across Jordan pieces whose overlaps have content zero (Jordan content is finitely additive when the overlap has content zero).

Proof

technique · reduction

1.1

Normalize at $a$ by the affine map $H (x) = a + A^{- 1} (g (x) - g (a)) .$ Choose $0 < q < ε$ and a slightly larger cube inside $U$ on which $∥ D H - I ∥_{2 \to 2} \leq q / \sqrt{n}$ . The mean-value bound in [L4] makes $H - I$ a $q$ -contraction in the sup norm, so $H$ is injective and bi-Lipschitz there. The derivative bound also makes every $D H$ invertible; the inverse function theorem in [L4] therefore makes $H$ a homeomorphism on a neighborhood of the smaller positive-radius cube $Q$ . Here $H (a) = a$ and $D H (a) = I$ ; continuity of $D g$ permits the stated choice inside any prescribed neighborhood.

L4given

2.1

If $E \subseteq Q$ is Jordan, the homeomorphism in step 1.1 gives $\partial H (E) = H (\partial E)$ . Compose $H$ on the larger cube with coordinatewise clamping onto that cube to obtain a global Lipschitz map. Since $\partial E$ is null, [L4] makes $H (\partial E)$ null and hence makes $H (E)$ Jordan.

L4step 1.1

3.1

Refine inner and outer figures from [L3] into finite unions of sufficiently small, interior-disjoint cubes $P \subseteq E \subseteq R$ with arbitrarily small content gap. After translating at each cube centre, [L2] sandwiches its $H$ -image between cubes with factors $(1 - q)^{n}$ and $(1 + q)^{n}$ . Step 2.1 makes those images Jordan, injectivity makes their interiors disjoint, and [L5] adds their contents. Letting the figure gap vanish gives the stronger bounds with $q$ ; since $q < ε$ , these imply the displayed bounds for $H (E)$ . Finally $g (E) = g (a) + A (H (E) - a)$ , so [L1] multiplies every content by $∣ \det A ∣ = ∣ \det D g (a) ∣$ .

L1L2L3L5step 1.1step 2.1∎

Depends on

Used by

Change of variables for an injective C¹ map on a compact Jordan set Theorem

Dependency tree · two levels

59 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Leibman, Multidimensional Real Analysis, Lemma 5.5.6 and Theorem 5.5.7 (standard reference, not scraped)

On a small cube, a C1C1 diffeomorphism distorts Jordan content by factors arbitrarily close to its linearized absolute determinant