AN ANALOGUE OF KOEBE’S THEOREM AND THE OPENNESS OF A LIMIT MAP IN ONE CLASS

Let us recall the formulation of the classical Koebe theorem, see, for example, [CG, Theorem 1.3].

Theorem A. Let $f:{\mathbb{D}}\rightarrow{\mathbb{C}}$ be an univalent analytic function such that $f(0)=0$ and $f^{\,\prime}(0)=1.$ Then the image of $f$ covers the open disk centered at $0$ of radius one-quarter, that is, $f({\mathbb{D}})\supset B(0,1/4).$

The main fact contained in the paper is the statement that something similar has been done for a much more general class of spatial mappings. Below $dm(x)$ denotes the element of the Lebesgue measure in ${\mathbb{R}}^{n}.$ Everywhere further the boundary $\partial A$ of the set $A$ and the closure $\overline{A}$ should be understood in the sense of the extended Euclidean space $\overline{{\mathbb{R}}^{n}}.$ Recall that, a Borel function $\rho:{\mathbb{R}}^{n}\,\rightarrow[0,\infty]$ is called admissible for the family $\Gamma$ of paths $\gamma$ in ${\mathbb{R}}^{n},$ if the relation

$$\int\limits_{\gamma}\rho(x)\,|dx|\geqslant 1$$

(1.1)

holds for all (locally rectifiable) paths $\gamma\in\Gamma.$ In this case, we write: $\rho\in{\rm adm}\,\Gamma.$ The modulus of $\Gamma$ is defined by the equality

$$M(\Gamma)=\inf\limits_{\rho\in\,{\rm adm}\,\Gamma}\int\limits_{{\mathbb{R}}^{n}}\rho^{n}(x)\,dm(x)\,.$$

(1.2)

Let $y_{0}\in{\mathbb{R}}^{n},$ $0<r_{1}<r_{2}<\infty$ and

$$A=A(y_{0},r_{1},r_{2})=\left\{y\,\in\,{\mathbb{R}}^{n}:r_{1}<|y-y_{0}|<r_{2}\right\}\,.$$

(1.3)

Given $x_{0}\in{\mathbb{R}}^{n},$ we put

$$B(x_{0},r)=\{x\in{\mathbb{R}}^{n}:|x-x_{0}|<r\}\,,\quad{\mathbb{B}}^{n}=B(0,1)\,,$$

$$S(x_{0},r)=\{x\,\in\,{\mathbb{R}}^{n}:|x-x_{0}|=r\}\,.$$

A mapping $f:D\rightarrow{\mathbb{R}}^{n}$ is called discrete if the pre-image $\{f^{-1}\left(y\right)\}$ of any point $y\,\in\,{\mathbb{R}}^{n}$ consists of isolated points, and open if the image of any open set $U\subset D$ is an open set in ${\mathbb{R}}^{n}.$

Given sets $E,$ $F\subset\overline{{\mathbb{R}}^{n}}$ and a domain $D\subset{\mathbb{R}}^{n}$ we denote by $\Gamma(E,F,D)$ the family of all paths $\gamma:[a,b]\rightarrow\overline{{\mathbb{R}}^{n}}$ such that $\gamma(a)\in E,\gamma(b)\in\,F$ and $\gamma(t)\in D$ for $t\in(a,b).$ Given a mapping $f:D\rightarrow{\mathbb{R}}^{n},$ a point $y_{0}\in\overline{f(D)}\setminus\{\infty\},$ and $0<r_{1}<r_{2}<r_{0}=\sup\limits_{y\in f(D)}|y-y_{0}|,$ we denote by $\Gamma_{f}(y_{0},r_{1},r_{2})$ a family of all paths $\gamma$ in $D$ such that $f(\gamma)\in\Gamma(S(y_{0},r_{1}),S(y_{0},r_{2}),A(y_{0},r_{1},r_{2})).$ Let $Q:{\mathbb{R}}^{n}\rightarrow[0,\infty]$ be a Lebesgue measurable function. We say that $f$ satisfies the inverse Poletsky inequality at a point $y_{0}\in\overline{f(D)}\setminus\{\infty\}$ if the relation

$$M(\Gamma_{f}(y_{0},r_{1},r_{2}))\leqslant\int\limits_{A(y_{0},r_{1},r_{2})\cap f(D)}Q(y)\cdot\eta^{n}(|y-y_{0}|)\,dm(y)$$

(1.4)

holds for any Lebesgue measurable function $\eta:(r_{1},r_{2})\rightarrow[0,\infty]$ such that

$$\int\limits_{r_{1}}^{r_{2}}\eta(r)\,dr\geqslant 1\,.$$

(1.5)

The definition of the relation (1.4) at the point $y_{0}=\infty$ may be given by the using of the inversion $\psi(y)=\frac{y}{|y|^{2}}$ at the origin.

Note that conformal mappings preserve the modulus of families of paths, so that we may write

$$M(\Gamma)=M(f(\Gamma))\,.$$

It is not difficult to see from this that conformal mappings from Koebe theorem satisfy the relation (1.4) with $Q\equiv 1$ for any function $\eta$ in (1.5).

Remark 1.1. It is known that the quasiregular mappings satisfy the inequality

$$M(\Gamma)\leqslant N(f,D)K_{O}(f)M(f(\Gamma))\,,$$

where $1\leqslant K_{O}(f)<\infty$ is some number, and $N(f,D)$ denotes the multiplicity function,

$$N(y,f,E)\,=\,{\rm card}\,\left\{x\in E:f(x)=y\right\}\,,$$

$$N(f,E)\,=\,\sup\limits_{y\in{\mathbb{R}}^{n}}\,N(y,f,E)\,,$$

(1.6)

see [MRV₁, Theorem 3.2]. There are also mappings in which the distortion of the modulus of families of paths is much more complex. Say, for homeomorphisms $f\in W^{1,n}_{\rm loc}$ such that $f^{\,-1}\in W^{1,n}_{\rm loc}$ we have the inequality

$$M(\Gamma)\leqslant\int\limits_{f(D)}K_{I}(y,f^{\,-1})\cdot\rho_{*}^{n}(y)\,dm(x)$$

(1.7)

for any $\rho_{*}\in{\rm adm}\,f(\Gamma)$ (see below), where

$$K_{I}(y,f^{\,-1})\quad=\sum\limits_{x\in f^{\,-1}(y)}K_{O}(x,f)\,,$$

(1.8)

$$K_{O}(x,f)\quad=\quad\left\{\begin{array}[]{rr}\frac{\|f^{\,\prime}(x)\|^{n}}{|J(x,f)|},&J(x,f)\neq 0,\\ 1,&f^{\,\prime}(x)=0,\\ \infty,&\text{otherwise}\end{array}\right.\,\,,$$

see [MRSY, Theorems 8.1, 8.6].

All of the above allows us to assert that relation (1.4) is satisfied by a fairly large number of mappings. In general, for practically all currently known classes, including conformal and quasiconformal mappings, quasiregular mappings, mappings with finite distortion, etc. such inequalities are satisfied.

Set

$$q_{y_{0}}(r)=\frac{1}{\omega_{n-1}r^{n-1}}\int\limits_{S(y_{0},r)}Q(y)\,d\mathcal{H}^{n-1}(y)\,,$$

(1.9)

and $\omega_{n-1}$ denotes the area of the unit sphere ${\mathbb{S}}^{n-1}$ in ${\mathbb{R}}^{n}.$

We say that a function ${\varphi}:D\rightarrow{\mathbb{R}}$ has a finite mean oscillation at a point $x_{0}\in D,$ write $\varphi\in FMO(x_{0}),$ if

$$\limsup\limits_{\varepsilon\rightarrow 0}\frac{1}{\Omega_{n}\varepsilon^{n}}\int\limits_{B(x_{0},\,\varepsilon)}|{\varphi}(x)-\overline{{\varphi}}_{\varepsilon}|\ dm(x)<\infty\,,$$

where $\overline{{\varphi}}_{\varepsilon}=\frac{1}{\Omega_{n}\varepsilon^{n}}\int\limits_{B(x_{0},\,\varepsilon)}{\varphi}(x)\,dm(x)$ and $\Omega_{n}$ is the volume of the unit ball ${\mathbb{B}}^{n}$ in ${\mathbb{R}}^{n}.$ We also say that a function ${\varphi}:D\rightarrow{\mathbb{R}}$ has a finite mean oscillation at $A\subset\overline{D},$ write ${\varphi}\in FMO(A),$ if ${\varphi}$ has a finite mean oscillation at any point $x_{0}\in A.$ Let $h$ be a chordal metric in $\overline{{\mathbb{R}}^{n}},$

$$h(x,\infty)=\frac{1}{\sqrt{1+{|x|}^{2}}}\,,$$

$$h(x,y)=\frac{|x-y|}{\sqrt{1+{|x|}^{2}}\sqrt{1+{|y|}^{2}}}\qquad x\neq\infty\neq y\,.$$

(1.10)

and let $h(E):=\sup\limits_{x,y\in E}\,h(x,y)$ be a chordal diameter of a set $E\subset\overline{{\mathbb{R}}^{n}}$ (see, e.g., [Va, Definition 12.1]).

Given a continuum $E\subset D,$ $\delta>0$ and a Lebesgue measurable function $Q:{\mathbb{R}}^{n}\rightarrow[0,\infty]$ we denote by $\mathfrak{F}_{E,\delta}(D)$ the family of all mapping $f:D\rightarrow{\mathbb{R}}^{n},$ $n\geqslant 2,$ satisfying relations (1.4)–(1.5) at any point $y_{0}\in\overline{{\mathbb{R}}^{n}}$ such that $h(f(E))\geqslant\delta.$ The following statement holds.

Theorem 1.1. Let $D$ be a domain in ${\mathbb{R}}^{n},$ $n\geqslant 2,$ and let $B(x_{0},\varepsilon_{1})\subset D$ for some $\varepsilon_{1}>0.$

Assume that, $Q\in L^{1}({\mathbb{R}}^{n})$ and, in addition, one of the following conditions hold:

1) $Q\in FMO(\overline{{\mathbb{R}}^{n}});$

2) for any $y_{0}\in\overline{{\mathbb{R}}^{n}}$ there is $\delta(y_{0})>0$ such that

$$\int\limits_{0}^{\delta(y_{0})}\frac{dt}{tq_{y_{0}}^{\frac{1}{n-1}}(t)}=\infty\,.$$

(1.11)

Then there is $r_{0}>0,$ which does not depend on $f,$ such that

$$f(B(x_{0},\varepsilon_{1}))\supset B(f(x_{0}),r_{0})\qquad\forall\,\,f\in\mathfrak{F}_{E,\delta}(D)\,.$$

Remark 1.2. The condition $Q\in FMO(\infty)$ of the condition (1.11) for $y_{0}=\infty$ must be understood as follows: these conditions hold for $y_{0}=\infty$ if and only if the function $\widetilde{Q}:=Q\left(\frac{y}{|y|^{2}}\right)$ satisfies similar conditions at the origin.

Note that the above analogue of Koebe’s theorem has an important application in the field of convergence of mappings. Recall that, a mapping $f:D\rightarrow{\mathbb{R}}^{n}$ is called a $K$-quasiregular mapping, if the following conditions hold:

1) $f\in W_{loc}^{1,n}(D),$

2) the Jacobian $J(x,f)$ of $f$ at $x\in D$ preserves the sign almost everywhere in $D,$

3) $\|f^{\,\prime}(x)\|^{n}\leqslant K\cdot|J(x,f)|$ for almost any $x\in D$ and some constant $K<\infty,$ where $\|f^{\,\prime}(x)\|\,=\,\max\limits_{h\in{\mathbb{R}}^{n}\backslash\{0\}}\frac{|f^{\,\prime}(x)h|}{|h|}\,,\quad J(x,f)=\det f^{\,\prime}(x),$ see e.g. [Re, Section 4, Ch. I], cf. [Ri, Definition 2.1, Ch. I]. As is known, the class of mappings with bounded distortion is closed under locally uniform convergence. In particular, the following statement is true (see, for example, [Re, Theorem 9.2.II]).

Theorem B. Let $f_{j}:D\rightarrow{\mathbb{R}}^{n},$ $n\geqslant 2,$ $j=1,2,\ldots,$ be a sequence of $K$-quasiregular mappings converging to some mapping $f:D\rightarrow{\mathbb{R}}^{n}$ as $j\rightarrow\infty$ locally uniformly in $D.$ Then either $f$ is $K$-quasiregular, of $f$ is a constant. In particular, in the first case $f$ is discrete and open (see [Re, Theorems 6.3.II and 6.4.II]).

As for the classes we are studying in (1.4)–(1.5), the following analogue of Theorem B is valid for them.

Theorem 1.2.   Let $D$ be a domain in ${\mathbb{R}}^{n},$ $n\geqslant 2.$ Let $f_{j}:D\rightarrow{\mathbb{R}}^{n},$ $n\geqslant 2,$ $j=1,2,\ldots,$ be a sequence of open discrete mappings satisfying the conditions (1.4)–(1.5) at any point $y_{0}\in\overline{{\mathbb{R}}^{n}}$ and converging to some mapping $f:D\rightarrow{\mathbb{R}}^{n}$ as $j\rightarrow\infty$ locally uniformly in $D.$ Assume that the conditions on the function $Q$ from Theorem 1 hold. Then either $f$ is a constant, or $f$ is light and open.

Remark 1.3. The lightness of the mapping $f$ in Theorem 1 was established earlier, see [Sev₁], cf. [Cr]. The goal of the paper is to obtain the openness of this mapping, which will follow from Theorem 1. Note that mappings that satisfy conditions (1.4)–(1.5) may not be open. For example, let $x=(x_{1},\ldots,x_{n}).$ We define $f$ as the identical mapping in the closed domain $\{x_{n}\geqslant 0\}$ and set $f(x)=(x_{1},\ldots,-x_{n})$ for $x_{n}<0.$ Observe that, the mapping $f$ satisfies conditions (1.4)–(1.5) for $Q(y)\equiv 2.$ Indeed, $f$ preserves the lengths of paths, is differentiable almost everywhere and has Luzin’s $N$ and $N^{\,-1}$-properties. Therefore, $f$ is a mapping with a finite length distortion (for the definition see [MRSY, section 8]). Now, $f$ satisfies (1.7) with $Q:=K_{I}(y,f^{\,-1})\quad=\sum\limits_{x\in f^{\,-1}(y)}K_{O}(x,f)\leqslant 1+1=2$ by [MRSY, Theorems 8.1, 8.6]. Therefore $f$ satisfies conditions (1.4)–(1.5) for $Q(y)\equiv 2,$ as well.

As for the discreteness of the limit mapping $f$ in Theorem 1, whether this mapping will be such is currently unknown.

The following statement was proved in [Na, Lemma 2.1].

Proposition 2.1.   The number $\delta_{n}(r)=\inf M(\Gamma(F,F_{*},\overline{{\mathbb{R}}^{n}})),$ where the infimum is taken over all continua $F$ and $F_{*}$ in $\overline{{\mathbb{R}}^{n}}$ with $h(F)\geqslant r$ and $h(F_{*})\geqslant r,$ is positive for each $r>0$ and zero for $r=0.$

The following statement also may be found in [Na, Theorem 4.1]).

Proposition 2.2.   Let $\mathfrak{F}$ be a collection of connected sets in a domain $D$ and let $\inf h(F)>0,$ $F\in\mathfrak{F}.$ Then $\inf\limits_{F\in\mathfrak{F}}M(\Gamma(F,A,D))>0$ either for each or for no continuum $A$ in $D.$

The following statement may be found in [Vu, Lemma 4.3].

Proposition 2.3.   Let $D$ be an open half space or an open ball in ${\mathbb{R}}^{n}$ and let $E$ and $F$ be subsets of $D.$ Then

$$M(\Gamma(E,F,D))\geqslant\frac{1}{2}\cdot M(\Gamma(E,F,\overline{{\mathbb{R}}^{n}}))\,.$$

For a domain $D\subset{\mathbb{R}}^{n},$ $n\geqslant 2,$ and a Lebesgue measurable function $Q:{\mathbb{R}}^{n}\rightarrow[0,\infty],$ $Q(y)\equiv 0$ for $y\in{\mathbb{R}}^{n}\setminus f(D),$ we denote by $\mathfrak{F}_{Q}(D)$ the family of all open discrete mappings $f:D\rightarrow{\mathbb{R}}^{n}$ such that relations (1.4)–(1.5) hold for each point $y_{0}\in f(D).$ The following result holds (see [SSD, Theorem 1.1]).

Proposition 2.4. Let $n\geqslant 2,$ and let $Q\in L^{1}({\mathbb{R}}^{n}).$ Then for any $x_{0}\in D$ and any $r_{0}>0$ such that $0<r_{0}<{\rm dist}(x_{0},\partial D)$ the inequality

$$|f(x)-f(x_{0})|\leqslant\frac{C_{n}\cdot(\|Q\|_{1})^{1/n}}{\log^{1/n}\left(1+\frac{r_{0}}{2|x-x_{0}|}\right)}$$

(2.1)

holds for any $x,y\in B(x_{0},r_{0})$ and $f\in\mathfrak{F}_{Q}(D),$ where $\|Q\|_{1}$ denotes the $L^{1}$-norm of $Q$ in ${\mathbb{R}}^{n},$ and $C_{n}>0$ is some constant depending only on $n.$ In particular, $\mathfrak{F}_{Q}(D)$ is equicontinuous in $D.$

Let $D\subset{\mathbb{R}}^{n},$ $f:D\rightarrow{\mathbb{R}}^{n}$ be a discrete open mapping, $\beta:[a,\,b)\rightarrow{\mathbb{R}}^{n}$ be a path, and $x\in\,f^{\,-1}(\beta(a)).$ A path $\alpha:[a,\,c)\rightarrow D$ is called a maximal $f$-lifting of $\beta$ starting at $x,$ if $(1)\quad\alpha(a)=x\,;$ $(2)\quad f\circ\alpha=\beta|_{[a,\,c)};$ $(3)$ for $c<c^{\prime}\leqslant b,$ there is no a path $\alpha^{\prime}:[a,\,c^{\prime})\rightarrow D$ such that $\alpha=\alpha^{\prime}|_{[a,\,c)}$ and $f\circ\alpha^{\,\prime}=\beta|_{[a,\,c^{\prime})}.$ If $\beta:[a,b)\rightarrow\overline{{\mathbb{R}}^{n}}$ is a path and if $C\subset\overline{{\mathbb{R}}^{n}},$ we say that $\beta\rightarrow C$ as $t\rightarrow b,$ if the spherical distance $h(\beta(t),C)\rightarrow 0$ as $t\rightarrow b$ (see [MRV₂, section 3.11]), where $h(\beta(t),C)=\inf\limits_{x\in C}h(\beta(t),x).$ The following assertion holds (see [MRV₂, Lemma 3.12]).

Proposition 2.5. Let $f:D\rightarrow{\mathbb{R}}^{n},$ $n\geqslant 2,$ be an open discrete mapping, let $x_{0}\in D,$ and let $\beta:[a,\,b)\rightarrow{\mathbb{R}}^{n}$ be a path such that $\beta(a)=f(x_{0})$ and such that either $\lim\limits_{t\rightarrow b}\beta(t)$ exists, or $\beta(t)\rightarrow\partial f(D)$ as $t\rightarrow b.$ Then $\beta$ has a maximal $f$-lifting $\alpha:[a,\,c)\rightarrow D$ starting at $x_{0}.$ If $\alpha(t)\rightarrow x_{1}\in D$ as $t\rightarrow c,$ then $c=b$ and $f(x_{1})=\lim\limits_{t\rightarrow b}\beta(t).$ Otherwise $\alpha(t)\rightarrow\partial D$ as $t\rightarrow c.$

The following statement may be found in [Sev₂, Lemma 1.3].

Proposition 2.6.   Let $Q:{\mathbb{R}}^{n}\rightarrow[0,\infty],$ $n\geqslant 2,$ be a Lebesgue measurable function and let $x_{0}\in{\mathbb{R}}^{n}.$ Assume that either of the following conditions holds

Lemma 3.1.   Let $D$ be a domain in ${\mathbb{R}}^{n},$ let $x_{0}\in D,$ let $A$ be a (non-degenerate) continuum in $D,$ and let $\varepsilon_{1}>0$ be such that $B(x_{0},\varepsilon_{1})\subset D.$ Let $r>0$ and let $C_{j},$ $j=1,2,\ldots,$ be a sequence of continua in $B(x_{0},\varepsilon_{1})$ such that $h(C_{j})\geqslant r,$ $h(C_{j})=\sup\limits_{x,y\in C_{j}}h(x,y).$ Then there is $R_{0}>0$ such that

$$M(\Gamma(C_{j},A,D))\geqslant R_{0}\qquad\forall\,\,j\in{\mathbb{N}}\,.$$

Proof.   Let $A_{1}$ be an arbitrary (non-degenerate) continuum in $B(x_{0},\varepsilon_{1}).$ By Proposition 2, there is $R_{*}>0$ such that $M(\Gamma(C_{j},A_{1},\overline{{\mathbb{R}}^{n}}))\geqslant R_{*}$ for any $j\in{\mathbb{N}}.$ Now, by Proposition 2 $M(\Gamma(C_{j},A_{1},B(x_{0},\varepsilon_{1})))\geqslant\frac{1}{2}\cdot M(\Gamma(C_{j},A_{1},\overline{{\mathbb{R}}^{n}}))\geqslant R_{*}/2.$ Now $M(\Gamma(C_{j},A_{1},D))\geqslant R_{*}/2$ for any $j\in{\mathbb{N}}.$ Finally, $M(\Gamma(C_{j},A,D))\geqslant R_{*}/2$ for any $j\in{\mathbb{N}}$ by Proposition 2. For the completeness of the proof, we may put $R_{0}:=R_{*}.$ $\Box$

Lemma 3.2. Let $D$ be a domain in ${\mathbb{R}}^{n},$ $n\geqslant 2,$ let $A$ be a set in $D,$ and let $B(x_{0},\varepsilon_{1})\subset A$ for some $\varepsilon_{1}>0.$

Assume that, $Q\in L^{1}({\mathbb{R}}^{n})$ and, in addition, for any $y_{0}\in\overline{{\mathbb{R}}^{n}}$ there is $\varepsilon_{0}=\varepsilon_{0}(y_{0})>0$ and a Lebesgue measurable function $\psi:(0,\varepsilon_{0})\rightarrow[0,\infty]$ such that

$$I(\varepsilon,\varepsilon_{0}):=\int\limits_{\varepsilon}^{\varepsilon_{0}}\psi(t)\,dt<\infty\quad\forall\,\,\varepsilon\in(0,\varepsilon_{0})\,,\quad I(\varepsilon,\varepsilon_{0})\rightarrow\infty\quad\text{при}\quad\varepsilon\rightarrow 0\,,$$

(3.1)

and, in addition,

$$\int\limits_{A(y_{0},\varepsilon,\varepsilon_{0})}Q(y)\cdot\psi^{\,n}(|y-y_{0}|)\,dm(x)=o(I^{n}(\varepsilon,\varepsilon_{0}))\,,$$

(3.2)

as $\varepsilon\rightarrow 0,$ where $A(y_{0},\varepsilon,\varepsilon_{0})$ is defined in (1.3). Then there is $r_{0}>0,$ which does not depend on $f,$ such that

$$f(B(x_{0},\varepsilon_{1}))\supset B(f(x_{0}),r_{0})\qquad\forall\,\,f\in\mathfrak{F}_{E,\delta}(D)\,.$$

(3.3)

Remark 3.1. If $y_{0}=\infty,$ the relation (3.2) must be understood by the using the inversion $\psi(y)=\frac{y}{|y|^{2}}$ at the origin. In other words, instead of

$$\int\limits_{A(y_{0},\varepsilon,\varepsilon_{0})}Q(y)\cdot\psi^{\,n}(|y-y_{0}|)\,dm(y)=o(I^{n}(\varepsilon,\varepsilon_{0}))$$

we need to consider the condition

$$\int\limits_{A(0,\varepsilon,\varepsilon_{0})}Q\left(\frac{y}{|y|^{2}}\right)\cdot\psi^{\,n}(|y|)\,dm(y)=o(I^{n}(\varepsilon,\varepsilon_{0}))\,.$$

Proof of Lemma 3. Let us prove the lemma by contradiction. Assume that its conclusion is wrong, i.e., the relation (3.3) does not hold for any $r_{0}>0.$ Then for any $m\in{\mathbb{N}}$ there is $y_{m}\in{\mathbb{R}}^{n}$ and $f_{m}\in\mathfrak{F}_{E,\delta}(D)$ such that $|f_{m}(x_{0})-y_{m}|<1/m$ and $y_{m}\not\in f_{m}(B(x_{0},\varepsilon_{1})).$ Due to the compactness of $\overline{{\mathbb{R}}^{n}}$ we may consider that $y_{m}\rightarrow y_{0}$ as $m\rightarrow\infty,$ where $y_{0}\in\overline{{\mathbb{R}}^{n}}.$ Then also $f_{m}(x_{0})\rightarrow y_{0}$ as $m\rightarrow\infty.$ We may consider that $y_{0}\neq\infty.$

Since by the assumption $h(f_{m}(E))\geqslant\delta$ for any $m\in{\mathbb{N}}$ and $d(f_{m}(E))\geqslant h(f_{m}(E)),$ where $d(f_{m}(E))$ denotes the Euclidean diameter of $f_{m}(E),$ there is $\varepsilon_{2}>0$ such that

$$f_{m}(E)\setminus B(y_{0},\varepsilon_{2})\neq\varnothing,\qquad m=1,2,\ldots\,.$$

(3.4)

By (3.4), there is $w_{m}=f_{m}(z_{m})\in\overline{{\mathbb{R}}^{n}}\setminus B(y_{0},\varepsilon_{2}),$ where $z_{m}\in E.$ Since $E$ is a continuum, $\overline{{\mathbb{R}}^{n}}$ is a compactum and the set $\overline{{\mathbb{R}}^{n}}\setminus B(y_{0},\varepsilon_{2})$ is closed, we may consider that $z_{m}\rightarrow z_{0}\in E$ as $m\rightarrow\infty$ and $w_{m}\rightarrow w_{0}\in\overline{{\mathbb{R}}^{n}}\setminus B(y_{0},\varepsilon_{2}).$ Obviously, $w_{0}\neq y_{0}.$

By Proposition 2 the family $f_{m}$ is equicontinuous. Now, for any $\varepsilon>0$ there is $\delta=\delta(z_{0})>0$ such that $h(f_{m}(z_{0}),f_{m}(z))<\varepsilon$ whenever $|z-z_{0}|\leqslant\delta.$ Then, by the triangle inequality

$$h(f_{m}(z),w_{0})\leqslant h(f_{m}(z),f_{m}(z_{0}))+h(f_{m}(z_{0}),f_{m}(z_{m}))+h(f_{m}(z_{m}),w_{0})<3\varepsilon$$

(3.5)

for $|z-z_{0}|<\delta,$ some $M_{1}\in{\mathbb{N}}$ and all $m\geqslant M_{1}.$ We may consider that latter holds for any $m=1,2,\ldots.$ Since $w_{0}\in\overline{{\mathbb{R}}^{n}}\setminus B(y_{0},\varepsilon_{2}),$ we may choose $\varepsilon>0$ such that $\overline{B_{h}(w_{0},3\varepsilon)}\cap\overline{B(y_{0},\varepsilon_{2})}=\varnothing,$ where $B_{h}(w_{0},\varepsilon)=\{w\in\overline{{\mathbb{R}}^{n}}:h(w,w_{0})<\varepsilon\}.$ Then (3.5) implies that

$$f_{m}(E_{1})\cap\overline{B(y_{0},\varepsilon_{2})}=\varnothing,\qquad m=1,2,\ldots\,,$$

(3.6)

where $E_{1}:=\overline{B(z_{0},\delta)}.$

Join the points $y_{m}$ and $f_{m}(x_{0})$ by a segment $\beta_{m}:[0,1]\rightarrow\overline{B(f_{m}(x_{0}),1/m)}$ such that $\beta_{m}(0)=f_{m}(x_{0})$ and $\beta_{m}(0)=y_{m}.$ Let $\alpha_{m},$ $\alpha_{m}:[0,c_{m})\rightarrow B(x_{0},\varepsilon_{1}),$ be a maximal $f_{m}$-lifting of $\beta_{m}$ in $B(x_{0},\varepsilon_{1})$ starting at $x_{0}.$ The lifting $\alpha_{m}$ exists by Proposition 2. By the same Proposition either $\alpha_{m}(t)\rightarrow x_{1}\in B(x_{0},\varepsilon_{1})$ as $t\rightarrow c_{m}-0$ (in this case, $c_{m}=1$ and $f_{m}(x_{1})=y_{m}$), or $\alpha_{m}(t)\rightarrow S(x_{0},\varepsilon_{1})$ as $t\rightarrow c_{m}.$ Observe that, the first situation is excluded. Indeed, if $f_{m}(x_{1})=y_{m},$ then $y_{m}\in f_{m}(B(x_{0},\varepsilon_{1})),$ that contradicts the choice of $y_{m}.$ Thus, $\alpha_{m}(t)\rightarrow S(x_{0},\varepsilon_{1})$ as $t\rightarrow c_{m}.$ Observe that, $\overline{|\alpha_{m}|}$ is a continuum in $\overline{B(x_{0},\varepsilon_{1})}$ and $h(\overline{|\alpha_{m}|})\geqslant h(0,S(x_{0},\varepsilon_{1})).$ Let us to apply Lemma 3 for $A:=E_{1}:=B(z_{0},\delta),$ $C_{m}:=|\alpha_{m}|$ and $r=h(0,S(x_{0},\varepsilon_{1})).$ By this lemma we may find $R_{0}>0$ such that

$$M(\Gamma(\overline{|\alpha_{m}|},E_{1},D))\geqslant R_{0}\,,\qquad m=1,2,\ldots\,.$$

(3.7)

Let us show that the relation (3.7) contradicts the definition of the mapping $f_{m}$ in (1.4)–(1.5). Indeed, since $f_{m}(x_{0})\rightarrow y_{0}$ as $m\rightarrow\infty,$ for any $k\in{\mathbb{N}}$ there is a number $m_{k}\in{\mathbb{N}}$ such that

$$B(f_{m_{k}}(x_{0}),1/k)\subset B(y_{0},2^{\,-k})\,.$$

(3.8)

Since $|\beta_{m}|\in B(f_{m}(x_{0}),1/m),$ by (3.8) we obtain that

$$|\beta_{m_{k}}|\subset B(y_{0},2^{\,-k})\,,\qquad k=1,2,\ldots\,.$$

(3.9)

Let $k_{0}\in{\mathbb{N}}$ be such that $2^{\,-k}<\varepsilon_{2},$ where $\varepsilon_{2}$ is a number from (3.6), and let $\Gamma_{k}:=\Gamma(|\alpha_{m_{k}}|,E_{1},D).$ In this case, we observe that

$$f_{m_{k}}(\Gamma_{k})>\Gamma(S(y_{0},\varepsilon_{2}),S(y_{0},2^{\,-k}),A(y_{0},2^{\,-k},\varepsilon_{2}))\,,$$

(3.10)

see Figure 1 for the scheme of the proof.