Modulus of continuity of orientation preserving approximately differentiable homeomorphisms with a.e. negative Jacobian
aa r X i v : . [ m a t h . C A ] J a n MODULUS OF CONTINUITY OF ORIENTATION PRESERVINGAPPROXIMATELY DIFFERENTIABLE HOMEOMORPHISMS WITHA.E. NEGATIVE JACOBIAN
PAWE L GOLDSTEIN AND PIOTR HAJ LASZ
Abstract.
We construct an a.e. approximately differentiable homeomorphism of a unit n -dimensional cube onto itself which is orientation preserving, has the Lusin property (N)and has the Jacobian determinant negative a.e. Moreover, the homeomorphism togetherwith its inverse satisfy a rather general sub-Lipschitz condition, in particular it can bebi-H¨older continuous with an arbitrary exponent less than 1. Introduction
The main result.
It is well known that in the case of diffeomorphisms the sign of theJacobian J Φ carries topological information about the mapping Φ in the sense that it tellsus whether the diffeomorphism is orientation preserving or orientation reversing. In fact, itis not difficult to prove, using the notion of degree, that if Φ : Ω → Ω is a homeomorphismbetween domains in R n and if Φ is differentiable at points x , x ∈ Ω , then the Jacobianof Φ cannot be positive at x and negative at x , see [9, Theorem 5.22]. In particular,if a homeomorphism between Euclidean domains is differentiable a.e., then either J Φ ≥ J Φ ≤ f : E → R , defined on a measurable set E ⊂ R n , is said to be approximately differentiable at x ∈ E if there is a measurable set E x ⊂ E and a linearfunction L : R n → R such that x is a density point of E x andlim E x ∋ y → x | f ( y ) − f ( x ) − L ( y − x ) || y − x | = 0 . Mathematics Subject Classification.
Primary 46E35; Secondary 26B05, 26B10, 26B35, 74B20.
Key words and phrases. approximately differentiable homeomorphisms, orientation preserving, H¨oldercondition, approximation.P.G. was supported by FNP grant POMOST BIS/2012-6/3P.H. was supported by NSF grant DMS-1500647.
The mapping L is called the approximate derivative of f at x and it is denoted by ap Df ( x ).The approximate derivative is unique (if it exists). If a mapping Φ : E ⊂ R n → R n is approximately differentiable at x ∈ E , we define the approximate Jacobian as J Φ =det ap D Φ( x ).We will be interested in mappings that are approximately differentiable a.e.Diffeomorphisms map sets of measure zero to sets of measure zero, so it is natural toconsider a similar property for classes of more general mappings. We say that a mappingΦ : Ω → R n , defined on an open set Ω ⊂ R n , has the Lusin property (N) if it maps sets ofLebesgue measure zero to sets of Lebesgue measure zero.Thus homeomorphisms that are approximately differentiable a.e. and have the Lusinproperty are far reaching generalizations of diffeomorphisms and yet it turns out that forsuch mappings the classical change of variables formula is true [5, 6, 8]. Homeomorphismsthat belong to the Sobolev space W ,p are approximately differentiable a.e. but they do notnecessarily have the Lusin property. The Lusin property is a strong additional conditionthat plays an important role in geometric applications of Sobolev mappings.In this context Haj lasz, back in 2001, asked the following questions: (see [9, Section 5.4]and [10, p. 234]): Question 1.
Is it possible to construct a homeomorphism
Φ : (0 , n → R n which isapproximately differentiable a.e., has the Lusin property (N) and at the same time J Φ > on a set of positive measure and J Φ < on a set of positive measure? Question 2.
Is it possible to construct a homeomorphism
Φ : [0 , n → [0 , n which isapproximately differentiable a.e., has the Lusin property (N), equals to the identity on theboundary (and hence it is sense preserving in the topological sense), but J Φ < a.e.? Question 3.
Is it possible to construct a homeomorphism
Φ : (0 , n → R n of the Sobolevclass W ,p , ≤ p < n − , such that at the same time J Φ > on a set of positive measureand J Φ < on a set of positive measure? The answer to Question 1 is in the positive and it has been known to the authors since2001, but it has not been published until very recently. Namely, in the paper [7] theauthors answered in the positive both questions 1 and 2. The Question 3 has also beenanswered in a sequence of surprising and deep papers [2, 10, 11]. For further motivationfor the problems considered here we refer the reader to papers [2, 7, 10, 11], especially to[7], because the results proved here are strictly related to those in [7].The uniform metric in the space of homeomorphisms of the unit cube Q = [0 , n ontoitself is defined by(1.1) d (Φ , Ψ) = sup x ∈ Q | Φ( x ) − Ψ( x ) | + sup x ∈ Q | Φ − ( x ) − Ψ − ( x ) | . The main result of [7] reads as follows.
Theorem 1.1.
There exists an almost everywhere approximately differentiable homeomor-phism Φ of the cube Q = [0 , n onto itself, such that RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 3 (a) Φ | ∂Q = id , (b) Φ is measure preserving, (c) Φ is a limit, in the uniform metric d , of a sequence of measure preserving C ∞ -diffeomorphisms of Q that are identity on the boundary, (d) the approximate derivative of Φ satisfies (1.2) ap D Φ( x ) = . . . . . . ... ... . . . ... ... . . . . . . − a.e. in Q. Note that (b) implies the Lusin condition (N). The proof given in [7] does not giveany estimates for the modulus of continuity of the homeomorphism Φ. One of our mainconcerns in Theorem 1.1 were the conditions (b) and (c). Satisfying them required theuse of results of Dacorogna and Moser [4], which do not provide any reasonable estimatesfor the modulus of continuity. So what can we say about the regularity of Φ if we dropthe conditions (b) and (c)? While Φ cannot be Lipschitz continuous (because Lipschitzcontinuous functions are differentiable a.e.), it is natural to ask how close the modulus ofcontinuity of Φ can be to the Lipschitz one. For example, can Φ be H¨older continuous?A positive answer is given in the next result which is the main result of the paper.
Theorem 1.2.
Assume φ : [0 , ∞ ) → [0 , ∞ ) satisfies the following conditions: (1) φ is increasing, concave, continuous and φ (0) = 0 , (2) Z dsφ ( s ) < ∞ . (3) t t − α φ ( t ) is increasing on (0 , for some α ∈ (0 , .Then there exists a homeomorphism F of the cube Q = [0 , n onto itself such that • F | ∂Q = id , • F has the Lusin property (N), • F is approximately differentiable a.e., • the approximate Jacobian J F is negative a.e. in Q , • for any x, y ∈ Q we have | F ( x ) − F ( y ) | + | F − ( x ) − F − ( y ) | ≤ C ( n, φ ) φ ( | x − y | ) . Here C ( n, φ ) denotes a constant that depends on n and φ only.The function φ ( t ) = Ct β , β ∈ (0 ,
1) satisfies properties (1), (2) and (3), so both Φ andΦ − can be β -H¨older continuous at the same time. Also φ ( t ) = t log t satisfies (1) nearzero and it can be extended to [0 , ∞ ) in a way that all conditions (1), (2) and (3) aresatisfied. It is easy to see that in that case Φ and Φ − are H¨older continuous with anyexponent β <
1. In particular Φ and Φ − can be in the fractional Sobolev space W s,p forall 0 < s < < p < ∞ .Condition (2) is the main estimate describing the growth of the function φ , while (1)simply means that φ is a modulus of continuity and (3) is of technical nature. Condition PAWE L GOLDSTEIN AND PIOTR HAJ LASZ (3) implies that φ is a modulus that is at least H¨older with some positive exponent α .Since, in applications, we are interested in moduli very close to Lipschitz, the condition(3) is not restrictive.The proof of Theorem 1.2 involves lemmata of purely technical nature; they are collectedin Section 2. In order to give motivation for these technical arguments, we will describenow the main ideas of the proof of Theorem 1.2.1.2. How to prove the main result.
The main building block in the proof of Theo-rem 1.2 is the following result.
Proposition 1.3.
Assume φ : [0 , ∞ ) → [0 , ∞ ) satisfies (1) φ is increasing, concave, continuous and φ (0) = 0 , (2) Z dsφ ( s ) < ∞ .Then there exists an a.e. approximately differentiable homeomorphism Φ with the Lusinproperty (N) of the cube Q = [0 , n onto itself, and a compact set A in the interior of Q ,such that • Φ | ∂Q = id , • | A | > , • Φ is a reflection ( x , . . . , x n − , x n ) ( x , . . . , x n − , − x n ) on A , Φ( A ) = A , and Φ is a C ∞ -diffeomorphism outside A , • at almost all points of the set A (1.3) ap D Φ( x ) = . . . . . . ... ... . . . ... ... . . . . . . − , • for any x, y ∈ Q we have | Φ( x ) − Φ( y ) | + | Φ − ( x ) − Φ − ( y ) | ≤ C ( n, φ ) φ ( | x − y | ) . Note that we no longer require the technical condition (3) from Theorem 1.2.The set A in Proposition 1.3 is a Cantor set of positive measure defined in a standardway as the intersection of a nested family of cubes inside Q . The homeomorphism Φ isa diffeomorphism outside A and it is a mirror reflection when restricted to A . It easilyfollows that Φ has the Lusin property and that (1.3) is satisfied at all density points of A .The homeomorphism F from Theorem 1.2 is constructed as a limit of a sequence ofhomeomorphisms F k . The first homeomorphism F = Φ is defined as in Proposition 1.3.It has negative Jacobian in the set K = A . Note that in the complement of A , F is adiffeomorphism, so in a small neighborhood of each point in Q \ A , F is almost affine.Modifying F slightly we can change it to a homoeomorphism ˜ F that is affine in manysmall cubes { Q i } i in the complement of A . RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 5
A generic modification of a diffeomorphism in a way that it becomes affine in a neigh-borhood of a point is described in Lemma 2.8.Now, in each cube Q i we replace the affine map ˜ F by a suitably rescaled version of thehomeomorphism Φ from Proposition 1.3; it is rescaled in a way that it coincides with theaffine map ˜ F near the boundary of Q i . The resulting mapping F coincides with F in aneighborhood of A so it has negative Jacobian on A . Also in each cube Q i ⊂ Q \ A , F isa rescaled version of the homeomorphism from Proposition 1.3 and hence it has negativeJacobian on a compact set A i ⊂ Q i of positive measure. Thus F has negative Jacobian ona compact set K = A ∪ S i A i . Clearly K ⊂ K . The homeomorphism F is constructedin a way that it coincides with F near K = A ∪ S i A i (and hence it has negative Jacobianon K ) and in the complement of that set it has negative Jacobian on compact subsetsof many tiny cubes. Thus F has negative Jacobian on a compact set K that contains K . We construct homeomorphisms F , F , . . . and an increasing sequence of compact sets K , K , . . . in a similar manner. The construction guarantees that | Q \ S ∞ k =1 K k | = 0. Thesequence of of homeomorphisms { F k } is a Cauchy sequence with respect to the uniformmetric d . It is well known and easy to check that the space of homeomorphisms of Q ontoitself is complete with respect to the uniform metric d (see [7, Lemma 1.2]) so the sequence { F k } converges to a homeomorphism F . Since | Q \ S ∞ k =1 K k | = 0 and F k has negativeJacobian on K k , it follows that F has negative Jacobian almost everywhere.Each of the homeomorphisms F k satisfies the continuity condition(1.4) | F k ( x ) − F k ( y ) | + | F − k ( x ) − F − k ( y ) | ≤ C k φ ( | x − y | ) , because this is the estimate for Φ and F k contains many rescaled copies of Φ. The problemis that since the mappings became more and more complicated, the constant C k diverges to ∞ as k → ∞ and hence the homeomorphism F to which the homeomorphisms F k convergedoes not satisfy the desired estimate | F ( x ) − F ( y ) | + | F − ( x ) − F − ( y ) | ≤ Cφ ( | x − y | ).Note that in the argument above we never mentioned condition (3) from Theorem 1.2,since this condition is not needed for Proposition 1.3. Actually, condition (3) is used toovercome the problem with the blow up of the estimates. Namely, we have Lemma 1.4.
Assume that (1) φ : [0 , ∞ ) → [0 , ∞ ) is increasing, continuous, concave, φ (0) = 0 , (2) Z dtφ ( t ) < ∞ and (3) t t − α φ ( t ) is increasing on (0 , for some α ∈ (0 , .Then there exists ψ such that (a) ψ : [0 , ∞ ) → [0 , ∞ ) is increasing, continuous, concave, ψ (0) = 0 , (b) Z dtψ ( t ) < ∞ and (c) lim t → + ψ ( t ) φ ( t ) = 0 . PAWE L GOLDSTEIN AND PIOTR HAJ LASZ
The conditions (1), (2) and (3) are the same as in Theorem 1.2 and conditions (a), (b)are the same as conditions (1), (2) in Proposition 1.3. However, (c) means that ψ ( t ) ismuch smaller than φ ( t ) when t > | Φ( x ) − Φ( y ) | + | Φ − ( x ) − Φ − ( y ) | ≤ Cψ ( | x − y | ), and we use it in the definition of the sequence { F k } constructed above. Property (c) of ψ implies that | Φ( x ) − Φ( y ) | + | Φ − ( x ) − Φ − ( y ) | ≤ εφ ( | x − y | ), provided x and y are sufficiently close. That allows us to control the constantin the inequality (1.4) so well that we can take the constant C k on the right hand side of(1.4) to be independent of k . Passing to the limit shows that the limiting homeomorphism F also satisfies (1.4) with that constant.1.3. How to prove Proposition 1.3.
The set A from the statement of Proposition 1.3is a Cantor set constructed in a standard way. Let 1 = α > α > α > . . . be a decreasingsequence of positive numbers such that 2 α k +1 < α k . Let Q = Q be the initial cube ofedge-length α = 1. Let Q be the union of 2 n cubes inside Q , each of edge-length α .These 2 n cubes are ‘evenly’ distributed. Let Q be the union of 2 n · n = 2 n cubes inside Q , each of edge-length α . Namely, inside each of the 2 n cubes in Q we have 2 n smallercubes of edge-length α . Again, the cubes are ‘evenly’ distributed, see Figure 2. Thecondition 2 α k +1 < α k is necessary, as otherwise there would not be enough space for thecubes of the next generation. The Cantor set is then defined as A = T ∞ k =0 Q k . The volumeof Q k equals 2 kn α nk = (2 k α k ) n . Hence, in order for the Cantor set to have positive measure,we need lim k →∞ k α k > n cubes Q j , j = 1 , , . . . , n , of the first generation, that form the set Q , are placedin two layers: 2 n − cubes ( j = 1 , . . . , n − ) above the other 2 n − cubes ( j = 2 n − +1 , . . . , n ).We choose indices in such a way that the cube Q n − + j is right below the cube Q j .The homeomorphism Φ will be constructed as a limit of diffeomorphisms Φ k . Thediffeomorphism Φ of Q is identity near the boundary of Q and it exchanges the cubes Q j from the top layer with the cubes Q n − + j from the bottom layer. For each j =1 , , . . . , n − , the mapping Φ restricted to Q j is a translation of Q j onto Q n − + j and alsoΦ restricted to Q n − + j is a translation of Q n − + j onto Q j . This construction of Φ iscarefully described in Lemma 2.6.The diffeomorphism Φ coincides with Φ in Q \Q . Inside each cube Q j there is a familyof 2 n cubes of the second generation. Now, the diffeomorphism Φ exchanges cubes fromthe top layer of this family with the cubes from the bottom layer. The diffeomorphismΦ does this in every cube Q j , j = 1 , , . . . , n . The diffeomorphisms Φ k are defined in asimilar way. The sequence { Φ k } converges in the uniform metric to a homeomorphism Φ.The sequence of diffeomorphisms { Φ k } reverses the vertical order of cubes used in theconstruction of the Cantor set A . Hence, the limiting homeomorphism Φ restricted tothe Cantor set A is the reflection in the hyperplane x n = 1 /
2. Thus Φ is approximatelydifferentiable at the density points of A and the approximate derivative satisfies (1.3).Also, it follows from the construction of the sequence { Φ k } that Φ is a diffeomorphismoutside A , so it is differentiable there and the Lusin property of Φ easily follows. RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 7
We already pointed out that the numbers α k used in this construction must satisfy1 = α > α > . . . , 2 α k +1 < α k and lim k →∞ k α k >
0. That would be enough if we wantedto obtain a homeomorphism Φ with all properties listed in Proposition 1.3 but the lastone: | Φ( x ) − Φ( y ) | + | Φ − ( x ) − Φ − ( y ) | ≤ C ( n, φ ) φ ( | x − y | ). This condition requires amuch more careful choice of the sequence { α k } , since the numbers in the sequence mustbe related to the function φ . This is done in Lemma 2.2.The numbers β k = ( α k − − α k ) / k -th generationto the boundaries of cubes of the ( k − Structure of the paper.
Section 2 is of technical character: we recall the defini-tion and basic properties of the modulus of continuity there and then we prove severallemmata needed in the proofs of Proposition 1.3 and Theorem 1.2. In Section 3 we proveProposition 1.3 and in Section 4 we prove Theorem 1.2.2.
Auxiliary lemmata
This section starts with a definition and basic properties of the modulus of continuity, butthen it is focused on technical lemmata that will be used later in the proof of Proposition 1.3and Theorem 1.2. Some of these lemmata have already been mentioned in Introduction.The reader may skip this section, go directly to Section 3 and come back to the results ofSection 2 whenever necessary.We say that a continuous, non-decreasing and concave function φ : [0 , ∞ ) → [0 , ∞ ) suchthat φ (0) = 0 is a modulus of continuity of a function f : X → R defined on a metricspace ( X, d ) if(2.1) | f ( x ) − f ( y ) | ≤ φ ( d ( x, y )) for all x, y ∈ X .It is well known that every continuous function on a compact metric space has a modulusof continuity. The construction goes as follows. If f is constant, then we take φ ( t ) ≡ f is not constant. Let φ ( t ) = sup {| f ( x ) − f ( y ) | : d ( x, y ) ≤ t } . Sincethe function f is uniformly continuous and bounded, the function φ : [0 , ∞ ) → [0 , ∞ )is non-decreasing and lim t → + φ ( t ) = 0. Also, φ ( t ) is constant for t ≥ diam X , so M = sup t> φ ( t ) is positive and finite. Clearly, | f ( x ) − f ( y ) | ≤ φ ( d ( x, y )) for all x, y ∈ X .To obtain a modulus of continuity (as defined above), we define φ to be the least concavefunction greater than or equal to φ . Namely,(2.2) φ ( t ) = inf { αt + β : φ ( s ) ≤ αs + β for all s ∈ [0 , ∞ ) } . It is easy to see that φ is concave and φ ≥ φ , so (2.1) is satisfied and φ is nonnegative.Moreover, concavity implies continuity of φ on (0 , ∞ ), see [14, Theorem A, p.4]. Since α ≥ φ is non-decreasing. It remains to show that lim t → + φ ( t ) = 0. For0 < β < M define α = sup s> ( φ ( s ) − β ) /s . Note that α is positive and finite, because φ ( s ) − β < s . Clearly, φ ( s ) ≤ αs + β for all s >
0. Hence φ ( t ) ≤ αt + β ,so 0 ≤ lim sup t → + φ ( t ) ≤ β . Since β > t → + φ ( t ) = 0. PAWE L GOLDSTEIN AND PIOTR HAJ LASZ
The moduli φ ( t ) = Ct describe Lipschitz functions and more generally φ ( t ) = Ct α , α ∈ (0 , α -H¨older continuous functions. In this paper we are interested infunctions with modulus of continuity satisfying the following conditions:(1) φ : [0 , ∞ ) → [0 , ∞ ) is increasing, continuous, concave, φ (0) = 0,(2) Z dtφ ( t ) < ∞ .Here and in what follows, by an increasing function we mean a strictly increasing function.The Lipschitz modulus of continuity φ ( t ) = Ct does not satisfy the condition (2). However, φ ( t ) = Ct α , α ∈ (0 , φ that equals φ ( t ) = t log t near zero and has both properties (1) and (2). A function with this modulusof continuity is H¨older continuous with any exponent α <
1. Thus condition (2) meansthat φ is a sub-Lipschitz modulus of continuity and it can be pretty close to the Lipschitzone.In order to avoid confusion we adopt the rule that φ − ( t ) will always stand for the inversefunction and not for its reciprocal.Although the following observation will not be used in the paper, one should note that (2)is equivalent to φ − ( t ) /t satisfying the Dini condition. Indeed, according to Lemma 2.1(b)we have lim t → + t/φ ( t ) = 0. Since concave functions are locally Lipschitz, [14, Theo-rem A, p.4], integration by parts yields Z dtφ ( t ) = tφ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) + Z tφ ′ ( t ) dtφ ( t ) = 1 φ (1) + Z φ (1)0 φ − ( s ) /s dss . The last equality follows from the substitution s = φ ( t ). Also, for the integration by partsto be rigorous, we should integrate from ε to 1 and then let ε → + .Now we will present some technical lemmata related to functions φ satisfying (1) and(2). They will be needed later in the proof of Proposition 1.3. Since the remaining partof this section is of purely technical nature, the reader might want to skip it for now andreturn to it when necessary.In Lemmata 2.1, 2.2, 2.3 and 2.4 below we assume that φ : [0 , ∞ ) → [0 , ∞ ) is a givenfunction that satisfies conditions (1) and (2). Lemma 2.1. (a)
The function t tφ ( t ) is non-decreasing and (b) lim t → + tφ ( t ) = 0 . Proof.
By concavity of φ , if s > t > φ ( t ) = φ (cid:16) ts s + (cid:16) − ts (cid:17) (cid:17) ≥ ts φ ( s ) + (cid:16) − ts (cid:17) φ (0) = ts φ ( s ) , RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 9 which proves (a). Next, (a) implies that the limit in (b) exists; assume, to the contrary,that lim t → + t/φ ( t ) = c >
0. Then, for all t >
0, 1 /φ ( t ) ≥ c/t and the improper integral in(2) is divergent. This proves (b). (cid:3) Lemma 2.2.
Let
N > be such that (2.4) 12 N Z φ − (2 − N )0 dsφ ( s ) ! = 1 . For k = 0 , , , . . . set α k = 12 N + k Z φ − (2 − N − k )0 dsφ ( s ) ! . Then (a) α = 1 , (b) 2 α k +1 < α k , (c) lim k →∞ k α k = 2 − N > , (d) for all k we have α k ≤ − k and there exists K o such that (2.5) 12 N + k < α k < N + k − for all k ≥ K o .Proof. The existence of
N > N → ∞ and as N → + . The properties (a)–(d)follow immediately from the definition of α k and the fact that R ds/φ ( s ) < ∞ . (cid:3) Lemma 2.3.
For k ≥ define β k = 14 ( α k − − α k ) = 12 N + k +1 Z φ − (2 − ( N + k − ) φ − (2 − ( N + k ) ) dsφ ( s ) . Then (a) φ (2 β k ) < − ( N + k )+1 , (b) 2 − ( N + k ) ≤ φ (4 β k ) , (c) the sequence ( β k ) is decreasing and lim k →∞ β k = 0 .Proof. Since, by assumptions, the function 1 /φ ( s ) is decreasing, we have β k < N + k +1 (cid:0) φ − (2 − ( N + k − ) − φ − (2 − ( N + k ) ) (cid:1) N + k ≤ φ − (2 − ( N + k − ) , (2.6)which proves (a). Similarly, β k > N + k +1 (cid:0) φ − (2 − ( N + k − ) − φ − (2 − ( N + k ) ) (cid:1) N + k − = 14 (cid:0) φ − (2 − ( N + k − ) − φ − (2 − ( N + k ) ) (cid:1) + 14 φ − (2 − ( N + k ) ) . (2.7)The function φ − is convex, φ − (0) = 0, thus for any t > φ − (2 t ) ≥ φ − ( t ) . Therefore, the expression (cid:0) φ − (2 − ( N + k − ) − φ − (2 − ( N + k ) ) (cid:1) is non-negative and ulti-mately(2.9) β k ≥ φ − (2 − ( N + k ) ) , which proves (b). Now, we claim that the sequence ( β k ) is decreasing. Indeed, by (2.6), β k < (cid:0) φ − (2 − ( N + k − ) − φ − (2 − ( N + k ) ) (cid:1) = 12 N + k +1 φ − (2 − ( N + k − ) − φ − (2 − ( N + k ) )2 − ( N + k − − − ( N + k ) . Convexity of φ − implies that the difference quotient φ − ( x ) − φ − ( y ) x − y is a non-decreasing function of both x and y . Therefore β k < N + k +1 φ − (2 − ( N + k − ) − φ − (2 − ( N + k − )2 − ( N + k − − − ( N + k − = 14 ( φ − (2 − ( N + k − ) − φ − (2 − ( N + k − )) < β k − , where the last inequality follows from (2.7). Finally, by taking k → ∞ in (a) we see thatlim k →∞ β k = 0, because φ is increasing, continuous and φ (0) = 0. (cid:3) Lemma 2.4.
For k ≥ define λ k = α k − /β k . Then there exists a constant C = C ( φ ) such that for all k ≥ , (2.10) sup ℓ We start by proving (2.10) for all k > K o , where K o is given as in (d), Lemma 2.2.We have then(2.11) λ k = α k − β k ≥ − N − k +1 β k ≥ φ (2 β k ) β k = 2 φ (2 β k )2 β k and(2.12) λ k ≤ − N − k +2 β k ≤ φ (4 β k ) β k = 16 φ (4 β k )4 β k ≤ φ (2 β k )2 β k , where the last inequality follows from concavity of φ , see Lemma 2.1(a).Since both the sequence ( β ℓ ) and the function φ ( t ) /t are non-increasing, the sequence φ (2 β ℓ ) / (2 β ℓ ) is non-decreasing sosup ℓ Assume that (a) φ : [0 , ∞ ) → [0 , ∞ ) is increasing, continuous, concave, φ (0) = 0 , (b) Z dtφ ( t ) < ∞ and (c) t t − α φ ( t ) is increasing on (0 , for some α ∈ (0 , .Then there exists ψ such that (1) ψ : [0 , ∞ ) → [0 , ∞ ) is increasing, continuous, concave, ψ (0) = 0 , (2) Z dtψ ( t ) < ∞ and (3) lim t → + ψ ( t ) φ ( t ) = 0 .Proof. Set a k = φ − (2 − k ).By an application of the Maclaurin-Cauchy integral test, we have (cf. Figure 1, left) ∞ X k =1 k ( a k − a k +1 ) ≤ Z a dtφ ( t ) ≤ ∞ X k =1 k +1 ( a k − a k +1 ) . Thus the convergence of the improper integral R a dtφ ( t ) is equivalent to the convergence ofthe series P ∞ k =1 k +1 ( a k − a k +1 ).Denoting A k = 2 k +1 ( a k − a k +1 ) we have, by assumption (b), that P ∞ k =1 A k < ∞ .One easily checks that ( A k ) − is equal to the slope of the secant of graph of the function φ through ( a k +1 , − ( k +1) ) and ( a k , − k ) (cf. Figure 1, right). Thus, by concavity of φ , thesequence ( A k ) is non-increasing.Now, let A ′ k = vuut ∞ X ℓ = k A ℓ − vuut ∞ X ℓ = k +1 A ℓ = A k vuut ∞ X ℓ = k A ℓ + vuut ∞ X ℓ = k +1 A ℓ − Then(2.14) lim k →∞ A ′ k A k = lim k →∞ vuut ∞ X ℓ = k A ℓ + vuut ∞ X ℓ = k +1 A ℓ − = ∞ , by convergence of the series P A ℓ . k k +1 a k +1 a k φ ( t ) ty a k +1 a k φ ( t )2 − ( k +1) − k ty k +1 a k − a k +1 Figure 1. Also, ∞ X k =1 A ′ k = vuut ∞ X k =1 A k < ∞ . However, the sequence ( A ′ k ) need not be non-increasing.To correct that, let us set A ′′ k = min ℓ ≤ k A ′ ℓ . The sequence ( A ′′ k ) is, obviously, non-increasing, moreover A ′′ k ≤ A ′ k , thus the series P A ′′ k is convergent.Next, we prove that lim k →∞ A ′′ k /A k = ∞ . Assume ℓ ( k ) ≤ k is such that A ′′ k = A ′ ℓ ( k ) .Since A ′ k > A ′ k → 0, we have ∀ m ∈ N ∃ k o ∀ k>k o A ′ k < A ′ ℓ ( m ) = min { A ′ , . . . , A ′ m } . Hence for k > k o A ′ ℓ ( k ) = min { A ′ , . . . , A ′ k } ≤ A ′ k < A ′ ℓ ( m ) = min { A ′ , . . . , A ′ m } so ℓ ( k ) 6∈ { , . . . , m } and thus ℓ ( k ) > m . This shows that ℓ ( k ) → ∞ as k → ∞ . Now(2.14) yields A ′′ k A k = A ′ ℓ ( k ) A k ≥ A ′ ℓ ( k ) A ℓ ( k ) k →∞ −−−→ ∞ . Set b k = ∞ X ℓ = k A ′′ ℓ ℓ +1 .The sequence ( b k ) is decreasing and lim k →∞ b k = 0. Moreover, we have A ′′ k = 2 k +1 ( b k − b k +1 ).Define ψ as a piecewise-linear function, affine on all intervals [ b k +1 , b k ] and such that ψ ( b k ) = 2 − k .For t ∈ ( b k +1 , b k ), ψ ′ ( t ) = 1 /A ′′ k , thus ψ ′ is non-increasing and positive, which impliesthat ψ is concave and increasing. RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 13 The same argument through the Maclaurin-Cauchy integral test that, at the beginningof the proof, gave us the equivalence between the convergence of the series P ∞ k =1 A k andthe condition (b) for φ , allows us to conclude the condition (2) from the convergence ofthe series P ∞ k =1 A ′′ k . Also, ψ (0) = lim t → + ψ ( t ) = lim k →∞ ψ ( b k ) = lim k →∞ − k = 0 . We still need to prove (3), i.e. that lim t → + ψ ( t ) φ ( t ) = 0.By the Stolz-Ces`aro Theorem, [3, Theorem 2.7.1],lim k →∞ b k a k = lim k →∞ b k +1 − b k a k +1 − a k = lim k →∞ A ′′ k A k = ∞ . Note that the condition (c) implies that whenever k > kt < 1, we have φ ( kt ) >k α φ ( t ). Thus, for t ∈ ( b k +1 , b k ], where k is sufficiently large0 ≤ ψ ( t ) φ ( t ) ≤ ψ ( b k ) φ ( b k +1 ) ≤ − k ( b k +1 /a k +1 ) α φ ( a k +1 ) = 2 (cid:18) a k +1 b k +1 (cid:19) α k →∞ −−−→ . Therefore lim t → + ψ ( t ) φ ( t ) = 0. (cid:3) For a diffeomorphism Φ : Ω → R n of class C ∞ , Ω ⊂ R n , we define k D Φ k Ω = sup x ∈ Ω k D Φ( x ) k = sup x ∈ Ω sup | ξ | =1 | D Φ( x ) ξ | , k ( D Φ) − k Ω = sup x ∈ Ω k ( D Φ) − ( x ) k = sup x ∈ Ω sup | ξ | =1 (cid:12)(cid:12) ( D Φ( x )) − ξ (cid:12)(cid:12) . In order to obtain estimates on the modulus of continuity of the homeomorphism Φconstructed in Proposition 1.3, we need Lipschitz estimates for the building block of theconstruction: a ‘box-exchange’ diffeomorphism that switches top and bottom layers ofdyadic cubes of size 0 < α < within the unit cube. Lemma 2.6. Assume that Q , . . . , Q n are the closed n -dimensional cubes of edge-length < α < inside the unit cube Q = [0 , n , with dyadic (i.e. with all coordinates equal / or / ) centers. Let β = (1 − α ) / . Then there exists a smooth diffeomorphism F α : Q → Q such that (a) F α exchanges the ‘top layer’ cubes Q j with ‘bottom layer’ cubes Q n − + j in such away that the restriction of F α to a β/ -tubular neighborhood of each of Q j is atranslation; (b) F α is identity near ∂Q , (c) k DF α k Q + k D ( F − α ) k Q < C ( n ) /β for some constant C ( n ) dependent only on n .Proof. Suppose that we already constructed F α for α ∈ [ , ). If 0 < α < , we set F α = F . Since the dyadic cubes Q j of edge-length α are contained in the dyadic cubes of edge-length 1 / 4, the mapping F α = F will exchange the cubes Q j , so it will have the properties(a) and (b). The estimate (c) follows from the corresponding estimate for F : k DF α k Q + k D ( F − α ) k Q = k DF k Q + k D ( F − ) k Q < C ( n ) < C ( n ) β , since β < when 0 < α < .Thus it remains to show the construction of the mapping F α when ≤ α < .First, we sketch the construction of a smooth diffeomorphism G γ : [ − , n → [ − , n , γ ∈ (0 , ], such that(A) G γ is identity near ∂ [ − , n ,(B) G γ maps the cube [ − (1 − γ ) , − γ ] n to [ − , ] n (C) G γ acts on [ − (1 − . γ ) , − . γ ] n as a homogeneous affine scaling transformation(i.e. a homothety),(D) DG γ is bounded by C ( n ) /γ for some constant C ( n ) dependent only on n ,(E) DG − γ is bounded by C ( n ), independently of γ .Such a diffeomorphism can be constructed as follows: let D : S n − → R denote the distanceof a point ( r, ϑ ) ∈ ∂ [ − , n , given in radial coordinates, to the origin, as a function of thespherical coordinate ϑ : S n − ∋ ϑ ( r, ϑ ) ∈ ∂ [ − , n r ∈ R . Fix ε ≪ γ and let R ε : S n − → R approximate D smoothly from below:0 < D ( ϑ ) − R ε ( ϑ ) < ε for ϑ ∈ S n − .Set r ε = (1 − . γ + 2 ε ) R ε ; one immediately checks that r ε approximates (1 − . γ ) D from above. Note that R ε − r ε ≈ γ . Last, let ξ γ : [0 , → R be a smooth, increasingfunction such that ξ ( t ) = ( − γ ) for t ∈ [0 , ε ] , t ∈ [1 − ε, . We can find such ξ with ξ ′ bounded independently of γ ; note that ξ ( t ) ≥ − γ ) ≥ for all t . Then we can define G γ in radial coordinates as G γ ( r, ϑ ) = id for r > R ε ( ϑ )( rξ (cid:16) r − r ε ( ϑ ) R ε ( ϑ ) − r ε ( ϑ ) (cid:17) , ϑ ) for r ε ( ϑ ) ≤ r ≤ R ε ( ϑ )( r − γ ) , ϑ ) for r < r ε . Then G γ is a smooth diffeomorphism satisfying conditions (A) to (C). To obtain (D) and(E), note that G γ is a radial map, thus to find bounds on DG γ and DG − γ it is enough toestimate ∂ r | G γ | from above and below. ∂ r | G γ | = ξ (cid:18) r − r ε ( ϑ ) R ε ( ϑ ) − r ε ( ϑ ) (cid:19) + rξ ′ (cid:18) r − r ε ( ϑ ) R ε ( ϑ ) − r ε ( ϑ ) (cid:19) R ε ( ϑ ) − r ε ( ϑ ) , RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 15 and thus 12 ≤ ∂ r | G γ | ≤ C ( n ) γ for some constant C ( n ) depending only on n . This proves (D) and (E).Dividing Q into 2 n dyadic cubes of edge 1 / / G β = G − α to each of them we obtain a diffeomorphism H α of Q ontoitself that shrinks the 2 n cubes Q i of edge α to concentric cubes of edge 1 / ∂Q . Obviously, DH α and DH − α satisfy analogous estimates as those for DG α and DG − α .Let F denote a diffeomorphism of Q that exchanges the ‘top’ cubes of edge-length 1 / -tubular neighborhoods and thatis equal to identity near ∂Q . Then F α = H − α ◦ F ◦ H α satisfies the conditions (a) and (b).Moreover, k DF α k Q + k DF − α k Q ≤ k DH − α k Q · k DF k Q · k DH α k Q + k DH − α k Q · k DF − k Q · k DH α k Q ≤ C ( n ) /β. (cid:3) In the proof of Proposition 1.3 we shall use Lemma 2.6 to construct F α for α = α − k α k +1 ,where α k are as in Lemma 2.2. Since β = 14 (1 − α ) = 14 (cid:18) − α k +1 α k (cid:19) = β k +1 α k we have Corollary 2.7. Assume α k are as in Lemma 2.2. Then for any k = 0 , , , . . . there existsa smooth diffeomorphism F α , with α = α − k α k +1 , that satisfies conditions (a) and (b) ofLemma 2.6 and moreover k DF α − k α k +1 k Q + k D ( F − α − k α k +1 ) k Q < C ( n ) α k β k +1 . The next lemma is similar to [7, Lemma 3.8], but some estimates are new. Let us recallthe notation used in [7]. k D Φ k Ω = sup x ∈ Ω k D Φ( x ) k = sup x ∈ Ω sup | ξ | = | η | =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i,j =1 ∂ Φ ∂x i ∂x j ( x ) ξ i η j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . It follows from Taylor’s theorem that if Φ ∈ C ∞ and x ∈ B ( x o , r ) ⊂ Ω, then(2.15) | Φ( x ) − Φ( x o ) − D Φ( x o )( x − x o ) | ≤ k D Φ k Ω | x − x o | . Lemma 2.8. Let G : Ω → R n , Ω ⊂ R n , be a C ∞ -diffeomorphism such that M = k DG k Ω + k ( DG ) − k Ω + k D G k Ω < ∞ . Let E = B ( x o , r ) ⋐ Ω , B = B ( x o , r ) , D = B ( x o , r/ and let T ( x ) = G ( x o ) + DG ( x o )( x − x ) be the tangent map to G at x o . If (2.16) r < (cid:0) M + 1) ℓ (cid:1) − for some ℓ ∈ N , then (a) diam G ( B ) < − ℓ , (b) T ( D ) ⊂ G ( B ) ⊂ T ( E ) , (c) there is a C ∞ -diffeomorphism ˜ G which coincides with G on Ω \ B ( x o , r/ andcoincides with T on B ( x o , r/ such that (d) the mapping ˜ G is bi-Lipschitz on B with the bi-Lipschitz constant Λ = 2 M , i.e. Λ − | x − y | ≤ | ˜ G ( x ) − ˜ G ( y ) | ≤ Λ | x − y | for all x, y ∈ B .Sketch of the proof. Arguments that are similar to those that appear in the proof of Lemma 3.8in [7] will be sketched only; for details we refer the reader to [7].Note that (a) follows immediately from the condition (2.16) and the bound k DG k Ω < M since | G ( x ) − G ( y ) | ≤ k DG k Ω | x − y | ≤ M · r < − ℓ for all x, y ∈ B .In what follows, we assume, for simplicity, that x o = 0, i.e. that the balls B , D and E arecentered at the origin.The mapping ˜ G in (c), interpolating between G and T is given as˜ G ( x ) = T ( x ) + φ ( | x | /r )( G ( x ) − T ( x )) = T ( x ) + L ( x ) , where φ ∈ C ∞ ( R , [0 , φ ( t ) = 0 for t ≤ / φ ( t ) = 1 for t ≥ / k φ ′ k ∞ ≤ 9. Clearly ˜ G coincides with G on Ω \ B ( x o , r/ 5) and coincides with T on B ( x o , r/ G is adiffeomorphism.Elementary calculations show that for x ∈ B we have k DL ( x ) k < M r , which in turnallows us, for x, y ∈ B , to estimate | ˜ G ( x ) − ˜ G ( y ) | from below:(2.17) | ˜ G ( x ) − ˜ G ( y ) | ≥ | DG (0)( x − y ) | . The inequality (2.17) shows that ˜ G is injective on B , which suffices to prove that G is ahomeomorphism. Taking in (2.17) y = x + τ v (for some arbitrary v ∈ R n and sufficientlysmall τ ) gives 12 | DG (0) τ v | ≤ | ˜ G ( x ) − ˜ G ( x + τ v ) | , thus | D ˜ G ( x ) v | = lim τ → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ G ( x + τ v ) − ˜ G ( x ) τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | DG (0) v | , and non-degeneracy of DG (0) implies non-degeneracy of D ˜ G ( x ), thus ˜ G is a diffeomor-phism. This completes the proof of (c).Since the diffeomorphisms G and ˜ G agree on the boundary of the ball B , G ( B ) = ˜ G ( B )so T ( D ) = ˜ G ( D ) ⊂ ˜ G ( B ) = G ( B ), which is the first inclusion in (b). RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 17 To prove the other inclusion, G ( B ) ⊂ T ( E ), we argue as follows. If x ∈ B , then by(2.15) | G ( x ) − T ( x ) | ≤ k D G k Ω | x | < M r . On the other hand, the distance between the ellipsoids T ( ∂E ) and T ( B ) is larger than M r (2.18) dist (cid:0) T ( ∂E ) , T ( B ) (cid:1) > M r so G ( B ) ∩ T ( ∂E ) = ∅ and hence G ( B ) ⊂ T ( E ). To prove (2.18), observe that the distancebetween the ellipsoids is minimized along the shortest semi-axis (this fact follows easilyfrom geometric arguments or from the Lagrange multiplier theorem), i.e. if x ∈ ∂B is suchthat T ( x ) ∈ T ( ∂B ) and T (2 x ) ∈ T ( ∂E ) are on a shortest semi-axis of the ellipsoid T ( ∂E ),then dist (cid:0) T ( ∂E ) , T ( B ) (cid:1) = | T (2 x ) − T ( x ) | = | DG (0) x | ≥ | x |k ( DG (0)) − k > rM > M r . Finally, using the estimate k DL ( x ) k ≤ M r and the fact that k DT ( x ) k = k DG (0) k ≤ M we immediately get k D ˜ G ( x ) k ≤ k DT ( x ) k + k DL ( x ) k ≤ (1 + 10 r ) M ≤ M. Hence the Lipschitz constant of ˜ G on B is bounded by 2 M . To get an estimate for theLipschitz constant of ˜ G − , we return to (2.17):2 M | ˜ G ( x ) − ˜ G ( y ) | ≥ k ( DG ) − k Ω · | DG (0)( x − y ) |≥ | DG (0) − DG (0)( x − y ) | = | x − y | , which completes the proof of (d) and hence that of the lemma. (cid:3) Proof of Proposition 1.3 The proof is a slight reworking of the construction in [7, Lemma 2.1]; the main differenceis that we no longer require Φ to be measure preserving, and in return we obtain estimateson the modulus of continuity of Φ and Φ − .Let the sequences ( α k ) and ( β k ) be defined as in Lemma 2.2 and Lemma 2.3.Let Q k , . . . , Q k n be the closed n -dimensional cubes inside the unit cube Q = [0 , n , ofedge-length α − k − α k < / 2, with dyadic (i.e. with all coordinates equal 1 / / 4) centers q , . . . , q n , q j = ( q j, , . . . , q j,n − , q j,n ), such that q n − + j = ( q j, , . . . , q j,n − , − q j,n ). Thatmeans the first 2 n − cubes are in the top layer and the last 2 n − are in the bottom layer,right below the corresponding cubes from the upper layer.Our construction is iterative. The starting point is the diffeomorphism Φ = F α − α ,constructed in Corollary 2.7. This diffeomorphism rigidly rearranges (translates) cubes Q j of the edge-length α − α = α inside the unit cube Q . Moreover, with each of the cubes Q j , Φ translates also its neighborhood consisting of all the points with distance less than β / The diffeomorphism Φ coincides with Φ on Q \ S n j =1 Q j , but in the interior of eachcube Q j , rearranged by the diffeomorphism Φ , Φ is a rescaled and translated version ofthe diffeomorphism F α − α . It rearranges 2 n cubes of the edge-length α · α − α = α .Since the diffeomorphism F α − α is identity near the boundary of the cube Q , the rescaledversions of it, applied to the cubes Q j , are identity near boundaries of these cubes and hencethe resulting mapping Φ is a smooth diffeomorphism. The diffeomorphism Φ coincideswith Φ outside the 2 n cubes of the second generation rearranged by Φ and it is a rescaledand translated version of the diffeomorphism F α − α inside each of the cubes rearrangedby the diffeomorphism Φ . It rearranges 2 n cubes of the edge-length α · α − α = α etc.The diffeomorphism Φ k rearranges 2 kn k -th generation cubes Q kj (together with theirsmall neighborhoods). Denoting the union of all these k -th generation cubes by Q k , weobtain a descending family of compact sets. Let A = ∞ \ k =1 Q k . By (c), Lemma 2.2, |Q k | = 2 kn α nk has positive limit, thus A is a Cantor set of positivemeasure.Thanks to the fact that at each step the subsequent modifications leading from Φ k toΦ k +1 happen only in the k -th generation cubes, of diameter √ nα k , the sequence Φ k isconvergent in the uniform metric in the space of homeomorphisms. Therefore, the limitmapping Φ is a homeomorphism.On the Cantor set A , Φ acts as a reflection ( x , . . . , x n − , x n ) ( x , . . . , x n − , − x n ),and outside A , Φ is a diffeomorphism (for each x ∈ Q \ A there exists k ∈ N such that Φrestricted to a small neighborhood of x coincides with Φ k ). Thus Φ is a.e. approximatelydifferentiable and its approximate derivative is equal to (1.3) in the density points of A .Also, Φ has the Lusin property.We need yet to prove the continuity estimates for Φ.Let us note the following observations on the construction of subsequent generations ofcubes in our example: • Each k -th generation cube Q kj has a well defined ‘ancestor’ cube A ℓ ( Q kj ) in gener-ation ℓ , for 0 ≤ ℓ ≤ k , i.e. for ℓ = 0 , , . . . , k there exists a unique ℓ -th generationcube A ℓ ( Q kj ) such that Q kj ⊂ A ℓ ( Q kj ). • Whenever ℓ < k , dist( Q kj , Q \ A ℓ ( Q kj )) ≥ β ℓ +1 + β ℓ +2 + · · · + β k . • Denote the union of the family of all ℓ -th generation cubes by Q ℓ , for ℓ = 0 , , . . . Let k ∈ N and fix ℓ < k . The mapping Φ k , restricted to ˘ Q ℓj := Q ℓj \ Q ℓ +1 , coincideswith F α − ℓ α ℓ +1 , translated and rescaled by a factor of α ℓ . Since the rescaling is bothin the domain and in the range by the same factor, Corollary 2.7 yields k D Φ k k ˘ Q ℓj = k DF α − ℓ α ℓ +1 k Q ≤ C ( n ) α ℓ β ℓ +1 = C ( n ) λ ℓ +1 . Moreover, by construction, Φ k in the set N ℓj := { x ∈ Q : 0 < dist( x, Q ℓj ) < β ℓ / } RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 19 Figure 2. In white areas Φ is either F α or a composition of a translationwith a properly rescaled mapping F α /α . In the shaded areas Φ is anisometry: translation or identity. α α α β β here Φ = F α , | D Φ | C ( n ) λ . Φ = rescaled F α /α , | D Φ | C ( n ) λ . is an isometry – a translation. Similarly, Φ k is a translation on the k -th generationcubes. Therefore on these sets k D Φ k k = 1 (cf. Figure 2).Next, we prove that, for any ℓ , the diffeomorphism Φ ℓ satisfies the modulus of continuityestimate: for any x, y ∈ Q ,(3.1) | Φ ℓ ( x ) − Φ ℓ ( y ) | ≤ C ( n, φ ) φ ( | x − y | ) , with the constant C = C ( n, φ ) independent on ℓ .We prove it by induction. Setting Φ o = id we may assume that (3.1) holds for ℓ = 0.Indeed, according to Lemma 2.1(a) we have | Φ o ( x ) − Φ o ( y ) | = | x − y | ≤ √ nφ ( √ n ) φ ( | x − y | ) . In the inductive step, assume that for some k ∈ N the estimate (3.1) holds for all ℓ < k .Fix x, y ∈ Q , x = y . If both x and y lie outside the ( k − Q k − ,then, by the inductive assumption, | Φ k ( x ) − Φ k ( y ) | = | Φ k − ( x ) − Φ k − ( y ) | ≤ C ( n, φ ) φ ( | x − y | ) , since Φ k and Φ k − coincide outside Q k − . Thus, in what follows, we assume that x ∈ Q k − j ,where Q k − j ⊂ Q k − is a ( k − β i ) is decreasing to 0, thus either | x − y | > β / 10, or there exists m ∈ N such that β m +1 / < | x − y | ≤ β m / | x − y | > β / 10, then | Φ k ( x ) − Φ k ( y ) | ≤ diam Q = √ n ≤ √ nφ ( β / φ ( | x − y | ) = C ( n, φ ) φ ( | x − y | ) . If β m +1 / < | x − y | ≤ β m / 10 and 1 < m ≤ k − 1, then | x − y | ≤ β m < β m + · · · + β k − ≤ dist( Q k − j , Q \ A m − ( Q k − j )) . This shows that x, y ∈ A m − ( Q k − j ) and(3.2) | Φ k ( x ) − Φ k ( y ) | ≤ diam(Φ k ( A m − ( Q k − j ))) = √ nα m − , because Φ k ( A m − ( Q k − j )) is again one of the ( m − β m +1 / < | x − y | ≤ β m / 10 and m ≥ k , thendist( y, Q k − j ) ≤ | x − y | ≤ β m ≤ β k < β k − , thus either y ∈ Q k − j , or y ∈ N k − j . We have, as observed at the beginning of the proof, k D Φ k k Q k − j ∪ N k − j ≤ max {k D Φ k k N k − j , k D Φ k k ˘ Q k − j , k D Φ k k Q k } = max { , C ( n ) λ k , } = C ( n ) λ k , Since x, y ∈ Q k − j ∪ N k − j , the mean value theorem yields | Φ k ( x ) − Φ k ( y ) | ≤ C ( n ) λ k | x − y | By Lemma 2.4, λ k ≤ C ( φ ) λ m , which gives(3.3) | Φ k ( x ) − Φ k ( y ) | ≤ C ( n, φ ) λ m β m 10 = C ( n, φ ) α m − . Now, Lemmata 2.2, 2.3 and inequality (2.3) yield α m − ≤ − ( m − = 2 N +2 − ( N +( m +1)) ≤ N +2 φ (4 β m +1 ) ≤ · N +2 φ ( β m +1 / ≤ · N +2 φ ( | x − y | ) . This, combined with the estimates in (3.2) and (3.3), proves the estimate (3.1).Note that the inverse mapping Φ − ℓ is constructed in exactly the same way as Φ ℓ ; theonly difference is that in Lemma 2.6, in place of the diffeomorphism F exchanging ‘top’ and‘bottom layer’ cubes of edge length , we use its inverse F − , which obviously possesses thesame properties as F (it is a diffeomorphism that is identity near ∂Q and it exchanges ‘top’and ‘bottom layer’ cubes of edge length , together with their -tubular neighborhoods).Therefore, an estimate analogous to (3.1) holds for Φ − ℓ as well (possibly with a differentconstant): | Φ − ℓ ( x ) − Φ − ℓ ( y ) | ≤ C ( n, φ ) φ ( | x − y | ) . RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 21 Passing to the limit in the uniform metric d in the space of homeomorphisms we see thatif Φ = lim ℓ →∞ Φ ℓ , then | Φ( x ) − Φ( y ) | ≤ C ( n, φ ) φ ( | x − y | )and | Φ − ( x ) − Φ − ( y ) | ≤ C ( n, φ ) φ ( | x − y | ) . ✷ Remark 3.1. By the definition of β k (cf. Lemma 2.3) one immediately sees that theconvergence condition (2) on φ in Proposition 1.3 is natural and necessary for our con-struction: (2) holds if and only if the series P k β k is convergent. Recall (Figure 2) that β k is the distance from the boundary of the ( k − k -th generationcube, thus 4 β = 1 − α < 1, 4 β + 8 β = 1 − α < β + 4 β + 8 β + · · · + 2 k β k = 12 (cid:0) − k α k (cid:1) < , thus (2) is necessary for the sequence α k to be well adapted for our construction.4. Proof of Theorem 1.2 Let φ satisfy conditions (1), (2) and (3) of Theorem 1.2 and let ψ be obtained from φ according to the construction given in the proof of Lemma 2.5. In particular, constantsdependent on the choice of ψ depend on φ only.The starting point of our iterative construction is the homeomorphism F = Φ, givenby Proposition 1.3, but with estimates dependent on ψ instead of φ . Then F is a.e.approximately differentiable in Q , it has the Lusin property, F = id near ∂Q and thereexists a compact set C = A of positive measure such that the Jacobian J F is negative(equal − 1) in almost every point of C ; F is a diffeomorphism outside C .Moreover, for any x, y ∈ Q , | F ( x ) − F ( y ) | + | F − ( x ) − F − ( y ) | ≤ C ( n, ψ ) ψ ( | x − y | ) ≤ C ( n, φ ) φ ( | x − y | ) , with the last inequality true for some C ( n, φ ) thanks to (3), Lemma 2.5. We can assumethat C ( n, φ ) ≥ F k : Q → Q satisfyinga) F k is equal to identity near ∂Q ,b) F k has the Lusin property,c) there exists a compact set C k ⊂ Q such that • | C k | > • for a.e. x ∈ C k , the homeomorphism F k is approximately differentiable at x and its Jacobian J F k ( x ) is negative, • F k is a diffeomorphism outside C k . d) for any x, y ∈ Q , | F k ( x ) − F k ( y ) | ≤ (cid:16) · · · + 12 k (cid:17) C ( n, φ ) φ ( | x − y | ) , | F − k ( x ) − F − k ( y ) | ≤ (cid:16) · · · + 12 k (cid:17) C ( n, φ ) φ ( | x − y | ) . In the inductive step we construct F k +1 by modifying F k in sufficiently small balls outside C k , enlarging the set of points at which the approximate Jacobian is negative. To this end,let us choose an open set Ω such that Ω ⋐ Q \ C k , | Ω | > | Q \ C k | and | F k (Ω) | > | Q \ F k ( C k ) | .Set M = 1 + k DF k k Ω + k ( DF k ) − k Ω + k D F k k Ω < ∞ . We fill Ω with a finite family of pairwise disjoint balls { B i } i ∈ I = { B ( x i , r i ) } i ∈ I , B i ⋐ Ω,with sufficiently small radii r i , i.e. r i < ρ , with ρ to be determined later, in such a waythat (cid:12)(cid:12) [ i B i (cid:12)(cid:12) > | Ω | and (cid:12)(cid:12) [ i F k ( B i ) (cid:12)(cid:12) > | F k (Ω) | . Assume ρ < (cid:0) M + 1) k +1 (cid:1) − . Then we can modify F k according to Lemma 2.8inside each of the balls B i , obtaining an approximately differentiable homeomorphism ˜ F k which coincides with F k on some neighborhoods of ∂Q and of C k and which is affine on eachof the balls D i , concentric with B i , but with radius r i / 2. Namely ˜ F k ( x ) = T i ( x ) = F k ( x i ) + DF k ( x i )( x − x i ) for x ∈ D i . Obviously, ˜ F k is an orientation preserving diffeomorphism in Q \ C k . In particular, the affine maps T i are orientation preserving, with J T i = J F k ( x i ) > n -dimensional cube Q i , with edges parallel to the coordinate directions,into each of the balls D i and denote by S i : Q i → Q the standard similarity (scaling +translation) transformation between Q i and the unit cube Q . We define F k +1 = ( ˜ F k in Q \ S i ∈ I Q i ˜ F k ◦ S − i ◦ Φ ◦ S i in each of the Q i . Then in each of the cubes Q i there is a positive measure Cantor set A i such that • S − i ◦ Φ ◦ S i | A i is a symmetry, • outside A i , S − i ◦ Φ ◦ S i is a diffeomorphism of Q i onto itself, equal to the identitynear ∂Q i .The homeomorphism F k +1 on A i is a composition of a symmetry (orientation-reversingaffine map with the Jacobian equal − 1) and the orientation preserving affine tangentmapping T i with the Jacobian equal J F k ( x i ) > 0. Thus J F k +1 is negative in all the densitypoints of A i (in fact it is constant on the set of density points of A i : for a.e. x ∈ A i it isequal to − J F k ( x i )).Note that for each i , | A i | / | Q i | = | A | / | Q | = | A | so | A i | = | A || Q i | , where A is the Cantorset constructed in Proposition 1.3. Note that | A | depends on the dimension n and the RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 23 choice of ψ and hence it depends on n and φ only. Thus (cid:12)(cid:12) [ i A i (cid:12)(cid:12) = X i | A i | = | A | X i | Q i | = C ( n, φ ) X i | B i |≥ C ( n, φ ) 23 | Ω | ≥ C ( n, φ ) 23 · | Q \ C k | = 12 C ( n, φ ) | Q \ C k | . (4.1)We define C k +1 = C k ∪ S i A i . It easily follows from (4.1) that(4.2) (cid:12)(cid:12)(cid:12) Q \ [ k C k (cid:12)(cid:12)(cid:12) = 0 . Clearly, the homeomorphism F k +1 has the corresponding properties a), b) and c). Thenext step is proving the continuity estimate d) for F k +1 and F − k +1 . The arguments arerepetitive, thus we provide the details in the most complex cases and sketch the remainingones.We first prove the estimate for the intermediate step ˜ F k . For points x, y ∈ Q we haveto consider several cases.Let x, y S i B i . Then we have, by assumption, | ˜ F k ( x ) − ˜ F k ( y ) | = | F k ( x ) − F k ( y ) | ≤ (cid:16) · · · + 12 k (cid:17) C ( n, φ ) φ ( | x − y | ) . Similarly, for any x, y S i F k ( B i ), | ˜ F − k ( x ) − ˜ F − k ( y ) | = | F − k ( x ) − F − k ( y ) | ≤ (cid:16) · · · + 12 k (cid:17) C ( n, φ ) φ ( | x − y | ) . The remaining cases when at least one of the points is in a ball B i are more difficult.By (d), Lemma 2.8, the mapping ˜ F k is bi-Lipschitz in each of the closed balls B i , withbi-Lipschitz constant Λ = 2 M .Assume that ρ is such that for t < ρ we have t/φ ( t ) < Λ − − k − (recall that by (b),Lemma 2.1, t/φ ( t ) → t → + ). Note that ρ depends on n , φ and k only.Let x, y ∈ B i . Then | x − y | < ρ < ρ , thus(4.3) | ˜ F k ( x ) − ˜ F k ( y ) | ≤ Λ | x − y | ≤ φ ( | x − y | )2 k +1 . In the same way we prove that if x, y ∈ F k ( B i ), then | x − y | < ρ and again | ˜ F − k ( x ) − ˜ F − k ( y ) | ≤ Λ | x − y | ≤ φ ( | x − y | )2 k +1 . Let x ∈ B i , y ∈ B j , i = j . We note that the segment [ x, y ] must intersect the boundariesof B i and B j ; let z ∈ [ x, y ] ∩ ∂B i and w ∈ [ x, y ] ∩ ∂B j . We have then ˜ F k ( z ) = F k ( z ) and ˜ F k ( w ) = F k ( w ); thus, by (4.3), the inductive assumption, and the fact that φ is increasing, | ˜ F k ( x ) − ˜ F k ( y ) | ≤ | ˜ F k ( x ) − ˜ F k ( z ) | + | ˜ F k ( z ) − ˜ F k ( w ) | + | ˜ F k ( w ) − ˜ F k ( y ) |≤ φ ( | x − z | )2 k +1 + | F k ( z ) − F k ( w ) | + φ ( | w − y | )2 k +1 ≤ (cid:16) · · · + 12 k (cid:17) C ( n, φ ) φ ( | x − y | ) + φ ( | x − y | )2 k . (4.4)The estimates for ˜ F − k when x ∈ F k ( B i ), y ∈ F k ( B j ), i = j , are done in exactly the samemanner.Let x ∈ B i and y S i B i . This case is settled in the same way: we decompose thesegment [ x, y ] into [ x, z ] ∪ [ z, y ], where z ∈ ∂B i and use the triangle inequality; also theestimates for ˜ F − k are done in exactly the same manner.Ultimately, we obtain that for any x, y ∈ Q (4.5) | ˜ F k ( x ) − ˜ F k ( y ) | ≤ (cid:16) · · · + 12 k (cid:17) C ( n, φ ) φ ( | x − y | ) + φ ( | x − y | )2 k , and the same estimate for ˜ F − k .Now, let us turn to F k +1 and F − k +1 .Outside the union of the cubes Q i we have that F k +1 coincides with ˜ F k so d) followsfrom (4.5) and the fact that C ( n, φ ) > x, y ∈ Q i we have, by Proposition 1.3, | F k +1 ( x ) − F k +1 ( y ) | = | ˜ F k ◦ S − i ◦ Φ ◦ S i ( x ) − ˜ F k ◦ S − i ◦ Φ ◦ S i ( y ) |≤ C ( n, ψ )Λ λ − ψ ( λ | x − y | ) ≤ C ( n, ψ )Λ ψ ( | x − y | ) , (4.6)where λ > Q i and the unit cube. We used here the factthat ˜ F k is Λ-Lipschitz on Q i . The last inequality follows from (a), Lemma 2.1. In the sameway we prove that whenever x, y ∈ F k ( Q i ), we have(4.7) | F − k +1 ( x ) − F − k +1 ( y ) | ≤ C ( n, ψ ) λ ψ ( λ Λ | x − y | ) ≤ C ( n, ψ )Λ ψ ( | x − y | ) . Assume, in addition to the previous restrictions on ρ , that ψ ( t ) φ ( t ) < C ( n, ψ )Λ2 k for t < ρ ,where C ( n, ψ ) is the same constant as the one in inequalities (4.6) and (4.7). It is easy tosee that we can find such ρ depending on n , φ and k only. Recall that if x, y ∈ Q i ⊂ B i or x, y ∈ F k ( Q i ) ⊂ F k ( B i ), then | x − y | < ρ so(4.8) | F k +1 ( x ) − F k +1 ( y ) | ≤ C ( n, ψ )Λ ψ ( | x − y | ) ≤ φ ( | x − y | )2 k and similarly | F − k +1 ( x ) − F − k +1 ( y ) | ≤ φ ( | x − y | )2 k . These estimates imply d). RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 25 Assume now that x ∈ Q i , y ∈ Q j . Then the segment [ x, y ] intersects ∂Q i and ∂Q j ; let z ∈ [ x, y ] ∩ ∂Q i and w ∈ [ x, y ] ∩ ∂Q j . Note that F k +1 ( z ) = ˜ F k ( z ) and F k +1 ( w ) = ˜ F k ( w ).Proceeding exactly as in (4.4), we use the triangle inequality, (4.5) and (4.8) to get | F k +1 ( x ) − F k +1 ( y ) |≤ φ ( | x − z | )2 k + h(cid:16) . . . + 12 k (cid:17) C ( n, φ ) φ ( | z − w | ) + φ ( | z − w | )2 k i + φ ( | w − y | )2 k ≤ (cid:16) . . . + 12 k +1 (cid:17) C ( n, φ ) φ ( | x − y | ) , because 3 / k < C ( n, φ ) / k +1 .The same arguments prove the above estimate in the case when x ∈ Q i , y S j Q j . Theestimate for the inverse function F − k +1 follows in exactly the same manner. This completesthe proof of the inequalities in d) for k + 1.We proved that for all x, y ∈ Q and all k (4.9) | F k ( x ) − F k ( y ) | + | F − k ( x ) − F − k ( y ) | ≤ C ( n, φ ) φ ( | x − y | ) . One can show as in [7] that the sequence { F k } converges in the uniform metric (1.1) to ahomeomorphism F that has all properties listed in Theorem 1.2, but the Lusin property(N). However, instead of referring to [7] we will use a straightforward argument showingconvergence of a subsequence of { F k } . We proved in (4.9) that both families { F k } and { F − k } are equicontinuous. Since the families are bounded, it follows from the Arzel`a-Ascoli theorem that subsequences converge uniformly F k i ⇒ F and F − k i ⇒ G . Sinceid = F k i ◦ F − k i ⇒ F ◦ G we conclude that F ◦ G = id so F is a homeomorphism and that F − k i ⇒ F − . Clearly F | ∂Q = id . Passing to the limit in (4.9) gives | F ( x ) − F ( y ) | + | F − ( x ) − F − ( y ) | ≤ C ( n, φ ) φ ( | x − y | ) . It follows from the construction that for m ≥ k , F m | C k = F k | C k so F | C k = F k | C k . Since onthe set C k the mapping F = F k has the Lusin property, it is approximately differentiable,and J F k = J F < C k , it follows from (4.2) that F is approximately differentiablea.e. with J F < F has the Lusin property on the set S k C k , and it remains to show that | F ( Q \ S k C k ) | = 0. Equivalently, we need to show that | Q \ F ( C k ) | → k → ∞ .Since C k +1 = C k ∪ S i A i , it suffices to show that there is a constant C > 0, depending on n and φ only, such that (cid:12)(cid:12)(cid:12) F (cid:16) [ i A i (cid:17)(cid:12)(cid:12)(cid:12) ≥ C | Q \ F ( C k ) | = C | Q \ F k ( C k ) | . Recall that (cid:12)(cid:12)(cid:12) [ i F k ( B i ) (cid:12)(cid:12)(cid:12) > | F k (Ω) | > | Q \ F k ( C k ) | , so it suffices to show that | F ( A i ) | ≥ C | F k ( B i ) | . Let E i = B ( x i , r i ). According to (b), Lemma 2.8, F k ( B i ) = ˜ F k ( B i ) ⊂ T i ( E i ). Since | T i ( E i ) | = 2 n | T i ( B i ) | , we get | F k ( B i ) | ≤ n | T i ( B i ) | .Observe also that F ( A i ) = T i ( A i ), so | F ( A i ) || T i ( B i ) | = | T i ( A i ) || T i ( B i ) | = | A i || B i | = C ( n ) | A i || Q i | = C ( n ) | A | = C ( n, φ )and hence | F ( A i ) | = C ( n, φ ) | T i ( B i ) | ≥ − n C ( n, φ ) | F k ( B i ) | . 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Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Re-sults in Mathematics and Related Areas (3)], 26. Springer–Verlag, Berlin, 1993.[14] Roberts, A. W., Varberg, D. E.: Convex Functions. Pure and Applied Mathematics, Vol. 57.Academic Press, New York-London, 1973.[15] ˇSver´ak, V.: Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988), 105–127.[16] Tak´acs, L.: An increasing continuous singular function. Amer. Math. Monthly 85 (1978), 35–37. RIENTATION PRESERVING HOMEOMORPHISMS WITH NEGATIVE JACOBIAN 27 Pawe l GoldsteinInstitute of MathematicsFaculty of Mathematics, Informatics and MechanicsUniversity of WarsawBanacha 202-097 Warsaw, Poland [email protected]