ON BEHAVIOR OF HOMEOMORPHISMS WITH INVERSE MODULUS CONDITIONS

On Behavior of Homeomorphisms With Inverse Modulus Conditions

E. SEVOST’YANOV, S. SKVORTSOV
July 1, 2019
Abstract

We consider some class of homeomorphisms of domains of Euclidean space, which are more general than quasiconformal mappings. For these homeomorphisms, we have obtained theorems on local behavior of it’s inverse mappings in a given domain. Under some additional conditions, we proved results about behavior of mappings mentioned above in the closure of the domain.

2010 Mathematics Subject Classification: Primary 30C65; Secondary 32U20, 31B15

1 Introduction

In Euclidean space, questions connected with the equicontinuity of quasiconformal mappings and some of their generalizations are relatively well studied (see., e.g., [1, Theorem 19.2], [2, Theorem 3.17] and [3, Lemma 3.12, Corollary 3.22]). The behavior of such classes is also investigated when this domain is closed (see., e.g., [4, Theorem 3.1] and [5, Theorem 3.1]). The passage to inverse mappings in the latter case does not present difficulties, since, as it is known, the quasiconformality of a direct mapping implies the quasiconformality of the mapping (moreover, the quasiconformality constant of the mappings is one and the same, see. e.g., [1, Corollary 13.3]; see. also [1, Theorem 34.3]). In other words, the study of mappings, inverse to quasiconformal, does not bring anything new in comparison with investigation of quasiconformal mappings.

The situation essentially changes if instead of quasiconformal mappings we consider some more general class of homeomorphisms. Let means modulus of curve family (see [1]) and corresponds to Lebesque measure in Suppose that mapping is defined in domain and it is satisfying

where is a certain (given) fixed function (see, e.g., [6]). Recall that if and only if

 ∫γρ(x)|dx|⩾1∀γ∈Γ.

In particular, all conformal and quasiconformal mappings satisfy (1.1), where function equals 1 or some constant, respectively (see, e.g., [7, Theorems 4.6 and 6.10]). Note that in case of particular (unbounded) function we, generally speaking, can not replace by in (1.1). (For this occasion, see the example 2, cited at the end of this work). The study of mappings the inverses of which satisfy the relation (1.1) is a separate topic for research. In this note we are interested in the local behavior of such mappings in the domain and also in

It is necessary to take into the early results of the first author [8], where mappings with similar conditions were also studied. The main result is contained in [8, Theorem 6.1] and it is proved under the condition that two points of the domain are fixed by mappings, that it is difficult to call an optimal constraint. In particular, among linear fractional automorphisms of the unit circle onto itself is at most one such mapping, in view of which the indicated condition turns out to be meaningless. Our main goal is to study analogous families of mappings with a rejection of any conditions normalization. As example 1 shows at the end of the paper, it essentially enriches the results obtained in the article from the point of view of applications.

Main definition and denotes used below can be found in monographs [1] and [9] and therefore omitted. Let are arbitrary sets. Further we denote the family of all path that connect and in i.e и for Recall that the domain is called locally connected at the point if for every neighborhood of a point there is a neighborhood of a point such that is connected. The domain is locally connected in the if is locally connected at every point The boundary of is called weakly flat at a point if for every and every neighborhood of the point there is a neighborhood of such that for all continua intersecting and The boundary of the domain is weakly flat, if it is weakly flat at every point of boundary of

For domains and arbitrary Lebesgue measurable function for denote the family of all mappings such that is homeomorphism of the domain onto satisfying(1.1). The following assertion is valid.

Theorem 1.1. Suppose that and are a compacts in If then the family is equicontinuous in

For the number domains and continuum and arbitrary Lebesgue measurable function for denote by the family of all mappings such that is a homeomorphism of the domain onto satisfying (1.1), wherein The following assertion is valid.

Theorem 1.2. Suppose that the domain is locally connected at all boundary points, and are compacts in and the domain has a weakly flat boundary. We also suppose that any path-connected component is non-degenerate continuum. If then each mapping extends by continuity to the mapping in addition, and family consisting of all extended mappings is equicontinuous in

2 Auxiliary information

First of all, we establish two elementary statements that play an important role in the proof of the main results. Let be an open, closed or half-open interval in As usual, for a curve suppose:

 |γ|={x∈Rn:∃t∈[a,b]:γ(t)=x},

wherein, is called carrier (image) of the curve We say that the curve lies in the domain if in addition, we will say that the curves and do not intersect if their carriers do not intersect. The curve is called Jordan arc, if is a homeomorphism on The following (almost obvious) assertion is valid.

Lemma 2.1.  Let be a domain in locally connected on its boundary. Then any two pairs of different points и can be joined by disjoint curves and so, that for all

Proof.   Notice, that the points of the domain are locally connected on the boundary and are accessible from within the domain by means of curves (see, e.g., [9, Proposition 13.2]). In this case, if we connect the points and by an arbitrary Jordan arc in the domain not passing through the points and (which is possible in view of the local connection of on the boundary and the transition from the curve to the broken line if it is necessary). Then does not divide the domain as a set of topological dimension 1 (see [10, Corollary 1.5.IV]), which ensures the existence of the desired curve Thus, in the case of the assertion of Lemma 2 is established.

Now let then again the points and does not divide the domain ([10, Corollary 1.5.IV]). In this case, you can also connect points and by a Jordan arc in that does not pass through the points and In view of the Antoine theorem (see [11, Theorem 4.3, § 4]) the domain can be mapped onto some domain by means of a flat homeomorphism so, that и is a segment in We also note that the boundary points of the domain are reachable from within by means of curves. In this way, we can connect points and in by a Jordan arc which lies entirely in except perhaps its end point

It remains to show that the curve can be chosen so that it does not intersect the segment In fact, let crosses and let and are, respectively, the largest and the smallest values for which Suppose also that

 J=J(s)=φ(a)+(φ(b)−φ(a))s,s∈[0,1]

is a parametrization of the interval Let and be such that and Suppose Let and is a unit vector, orthogonal to then the set

 Pε={x=φ(a)+x1e1+x2e2,x1∈(−ε,s2+ε),x2∈(−ε,ε)},ε>0,

is a rectangle containing where is a restriction of to a segment (see picture 1).

We choose that so that In view of [12, Theorem 1.I, ch. 5, §  46]) the curve crosses for some and Let and Since is a connected set, it is possible to connect points and of the curve Finally, we put

 α∗2(t)={α2(t),t∈[0,1]∖[T1,T2],α∗(t),t∈[T1,T2]

and Then connects and in and connects and in while and do not intersect, which should be established.

Above we introduced the concept of a weak plane of the boundary of the region, without mentioning, at the same time, internal points. The following lemma contains the assertion that at the indicated points the property of the <<weak plane>> always takes place.

Lemma 2.2.   Let be a domain in and Then for every and for for any neighborhood of the point there is a neighborhood of the same point such that for arbitrary continua intersecting and

Proof.   Let be an arbitrary neighborhood of Let‘s choose so that Let be a positive constant, defined in the relation (10.11) in [1], and the number is so small that Suppose Let be arbitrary continua intersecting and then also and intersecting and (see [12, Theorem 1.I, ch. 5, §  46]). The necessary conclusion follows on the basis of [1, par. 10.12], because the

 M(Γ(E,F,D))⩾cn⋅logε0ε>P.□

3 Proof of Theorem 1

We prove the theorem 1 by contradiction. Suppose, the family is not equicontinuous at some point in other words, there are and such that for any there exists an element and a homeomorphism for which

 |gm(ym)−gm(y0)|⩾ε0. (3.1)

We draw a line through and (see picture 2).

Note that this line for must intersect the domain in view of [12, Theorem 1.I, ch. 5, §  46]), since the domain is bounded; thus, there exists such that Without loss of generality we can assume that for all then the segment belongs to for all and In view of analogous considerations, there are and a segment such that и belongs to for all Put Since is a homeomorphism, for each fixed the limit sets and of mappings at the corresponding boundary points lie on (see [9, Proposition 13.5]). Consequently, there is a point such that As is compact, it can be assumed that the sequence for Similarly, there is a sequence such that and for

Let be the part of the interval enclosed between the points and and be the part of the interval enclosed between the points and By construction and by (3.1), Let then the function

 ρ(x)={1ε0,x∈D,0,x∉D

is admissible for the family since for an arbitrary (locally rectifiable) curve it is completed (where denotes the length of the curve ). Since by the hypothesis the mappings satisfy (1.1) we obtain:

 M(fm(Γm))⩽1εn0∫DQ(x)dm(x):=c<∞, (3.2)

as On the other hand, and for large in addition

 dist(fm(Pm),fm(Qm))⩽|ym−y0|→0,m→∞.

Then, in view of Lemma 2

 M(fm(Γm))=M(fm(Pm),fm(Qm),D′)→∞,m→∞,

which contradicts relation (3.2). This contradiction indicates that the assumption in (3.1) is erroneous, which completes the proof of the theorem.

4 On the behavior of mappings in the closure of a domain

Let us pass to the question of the global behavior of mappings. The following assertion indicates that for sufficiently good domains and mappings with condition (1.1) the image of a fixed continuum under these mappings can not approach the boundary of the corresponding domain as soon as the Euclidean of the diameter of this continuum is bounded from below (see also [1, Theorems 21.13 and 21.14]).

Lemma 4.1.   Suppose that the domain is locally path-connected on and are compact sets in has a weakly flat boundary, and there is no connected component of the boundary degenerating to a point. Let be a sequence of homeomorphisms of the domain onto the domain with the condition (1.1). Let there also be a continuum and a number such that for all Then there exists such that

 dist(fm(A),∂D′)>δ1>0∀m∈N.

Proof.   Suppose, the contrary situation, that for each there exists Without loss of generality we can assume that the sequence is monotonically increasing. By condition is compact, therefore is also compact as a closed subset of the compactum In addition, is compact as a continuous image of the compactum under the mapping Then there are and such that (see picture 3).

As since is compact, we can assume that then also

 xk→y0∈∂D′,k→∞.

Let be a connected component containing the point then, obviously is a continuum in Since has a weakly flat boundary, for each the mapping extends to a continuous mapping (see [9, Theorem 4.6]), furthermore , uniformly continuous on as a mapping that is continuous on a compactum. Then for every there is such that

 |¯¯¯gmk(x)−¯¯¯gmk(x0)|<ε∀x,x0∈¯¯¯¯¯¯¯D′,|x−x0|<δk,δk<1/k. (4.1)

Let be an arbitrary number with the condition

 ε<(1/2)⋅dist(∂D,A), (4.2)

where is a continuum from the condition of the lemma. For each fixed we consider the set

Note that is an open set containing in other words, is a neighborhood of the continuum In view of [13, Lemma 2.2] there exists a neighborhood of the continuum such that is connected. Without loss of generality, we can assume that is an open set, then is also linearly connected (see [9, Proposition 13.1]). Let then there exist such that Hence, we can choose sequences and such that and for We can assume that

 |zk−wk|>m0/2,∀k∈N. (4.3)

We connect consecutively the points and of the curve in (this is possible, since is path-connected). Let be, as usual, the carrier (image) of the curve in Then is a compact set in Let then there is We will fix Because the then is an interior point of the domain so we have the right to write instead of for the indicated In this case, from (4.1) and (4.2), in view of the triangle inequality, for large we obtain:

 |gmk(x)−ω|⩾|ω−¯¯¯gmk(x0)|−|¯¯¯gmk(x0)−gmk(x)|⩾
 ⩾dist(∂D,A)−(1/2)⋅dist(∂D,A)=(1/2)⋅dist(∂D,A)>ε. (4.4)

Passing to (4.4) to over all and all we obtain:

 dist(gmk(|γk|),A)>ε,∀k=1,2,…. (4.5)

In view of (4.5) the length of an arbitrary curve joining compacta and in not less than Put then the function for and for is admissible for since for (where denotes the length of the curve ). By the definition of mappings in (1.1), we have:

 M(fmk(Γk))⩽1εn∫DQ(x)dm(x)=c=c(ε,Q)<∞, (4.6)

Since by hypothesis

We now show that we arrive at a contradiction with (4.6) in view of the weak boundary plane We choose at the point the ball where and – is a number from the condition of the lemma and Notice, that for sufficiently large because the и for In view of the same considerations As and are continua, then

 fmk(A)∩∂U≠∅,|γk|∩∂U≠∅, (4.7)

see [12, Theorem 1.I, гл. 5, §  46]. For a fixed let is a neighborhood of the point corresponding to the definition of a weakly flat boundary, that is, such that for any continua with condition and is satisfied the inequality

 M(Γ(E,F,D′))>P. (4.8)

We note that for sufficiently large

 fmk(A)∩∂V≠∅,|γk|∩∂V≠∅. (4.9)

Indeed where for therefore for large Besides and, since, in view of (4.3), then Then (see [12, Theorem 1.I, ch. 5, §  46]). Similarly, and, since by hypothesis, then In view of [12, Theorem 1.I, ch. 5, §  46] we have: The relations in (4.9) are established.

Thus, according to (4.7), (4.8) and (4.9), we get that

 M(Γ(fmk(A),|γk|,D′))>P. (4.10)

Notice, that so that inequality (4.10) can be rewritten in the form

 M(Γ(fmk(A),gmk(|γk|),D))=M(fmk(Γk))>P,

which contradicts inequality (4.6). The resulting contradiction indicates the incorrectness of the original assumption The lemma is proved.

Proof of Theorem 1. Since has a weakly flat boundary, each extends to a continuous mapping (see [9, Theorem 4.6]).

We verify equality In fact, by definition It remains to show the converse inclusion Let then we show that If then either or If then there is nothing to prove, since by hypothesis Now let then there be and such that and for Since is compact, we can assume that for Since is a homeomorphism, then Since is continuous in However, in this case, since and Hence, The inclusion is proved and, hence, as required.

Equicontinuity of curve family at interior points is the result of the theorem 1. It remains to show that this family is equicontinuous at the boundary points. We carry out the proof by contradiction. Suppose we find a point a number and sequences for and such that

 |¯¯¯gm(zm)−¯¯¯gm(z0)|⩾ε0,m=1,2,…. (4.11)

Put Since extends by continuity to the boundary of we can assume that and, hence, In addition, there is one more sequence for such that for Since is compact, we can assume that the sequences and are convergent for Let and for By continuity of the modulus from (4.11) it follows that moreover, since the homeomorphisms preserve the boundary, Let and be arbitrary distinct points of the continuum none of which coincide with с By Lemma 2 we can join points and by the path and points and by the curve such that for all