Abstract
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MATHEMATICS
V. P. OREVKOV
A CONSTRUCTIVE MAPPING OF THE SQUARE INTO ITSELF THAT MOVES EACH CONSTRUCTIVE POINT
(Presented by Academician P. S. Novikov on 2 IV 1963)
In the present communication we shall use the notions of a constructive operator from one constructive metric space into another, a continuous constructive operator, and a uniformly continuous constructive operator. Definitions of these notions may be found in [1]. Following [1], we denote by $\mathcal{E}_{2,2}$ the space of pairs of constructive real numbers*, i.e., words of the form $x\sigma y$, where $x$ and $y$ are constructive real numbers. In what follows, pairs of constructive real numbers will be called points.
A lattice will mean any word of the form $P \square x\Delta y$, where $P$ is a system of pairwise disjoint rational segments (see [2]) and $x\Delta y$ is a segment containing the system $P$. We shall say that the point $x\sigma y$ belongs to the lattice $P \square x_1\Delta y_1$, and shall write $x\sigma y \in P \square x_1\Delta y_1$, if
\[ (x \in P \ \&\ y \in x_1\Delta y_1) \quad \text{or}^{**} \quad (y \in P \ \&\ x \in x_1\Delta y_1). \]
The lattice $x\Delta y \square x\Delta y$ will be called a square with vertices at the points $x\sigma x$, $x\sigma y$, $y\sigma x$, $y\sigma y$ and will be denoted by $x t y$. We shall say that the point $x_1\sigma y_1$ lies on the boundary of the square $x t y$ if $x_1\sigma y_1 \in x t y$ and
\[ x_1 = x,\quad \text{or } x_1 = y,\quad \text{or } y_1 = x,\quad \text{or } y_1 = y. \]
The inclusion relation for lattices is defined in the natural way and is denoted by the sign $\subseteq$.
Definition 1. A constructive operator $F$ from $\mathcal{E}_{2,2}$ into $\mathcal{E}_{2,2}$ will be called a retraction of the square $x t y$ onto its boundary if the following conditions are satisfied:
1) whatever the point $x_1\sigma y_1$ belonging to the square $x t y$, $F$ is applicable to the point $x_1\sigma y_1$ and $F(x_1\sigma y_1)$ belongs to the boundary of the square $x t y$;
2) whatever the point $x_1\sigma y_1$ belonging to the boundary of the square $x t y$,
\[ F(x_1\sigma y_1)=x_1\sigma y_1. \]
Definition 2. A constructive operator $F$ from $\mathcal{E}_{2,2}$ into $\mathcal{E}_{2,2}$ will be called a mapping of the square $x t y$ into itself without a fixed point if, whatever the point $x_1\sigma y_1$ belonging to $x t y$, $F$ is applicable to the point $x_1\sigma y_1$ and
\[ (F(x_1\sigma y_1)\in x t y \ \&\ F(x_1\sigma y_1)\ne x_1\sigma y_1). \]
Theorem 1. One can construct a continuous retraction of the square $-1 t 1$ onto its boundary.
* Here and below, by a constructive real number we mean a real FR-number (a real duplex) [1].
** Following [1], by the expression “$A$ or $B$” we shall mean the formula $\neg\neg(A\vee B)$.
Theorem 2. One can construct a continuous mapping \(F\) of the square \(-1\tau 1\) into itself, without a fixed point, such that for any point \(x\sigma y\) belonging to the square \(-1\tau 1\),
\[ \rho\bigl(F(x\sigma y)\,\square\, x\sigma y\bigr)\geqslant {}^{1}/_{8}, \tag{1} \]
where \(\rho\) is the metric function in the space \(\overline{\mathcal E}_{2,2}\).
Theorem 3. One can construct a uniformly continuous mapping of the square \(-1\tau 1\) into itself, without a fixed point.
Let us outline the plan of proof of these theorems. First, algorithms \(\mathfrak A\) and \(\mathfrak B\) are constructed such that:
1) the algorithm \(\mathfrak A\) transforms any natural number into a system of pairwise disjoint rational segments contained in the segment \(-{}^{1}/_{2}\Delta {}^{1}/_{2}\) and consisting of an odd number of segments;
2) for any natural number \(n\),
\[ \mathfrak A(n+1)\,\square\, -1\Delta 1 \subseteq \mathfrak A(n)\,\square\, -1\Delta 1 \subseteq -1\tau 1; \]
3) the algorithm \(\mathfrak B\) transforms any constructive real number \(x\) into such a natural number that
\[ \neg\bigl(x\in \mathfrak A(\mathfrak B(x))\bigr). \]
The construction of the algorithms \(\mathfrak A\) and \(\mathfrak B\) differs only slightly from the construction of analogous algorithms in Theorems 4.1 and 4.2 of [2].
The most difficult part of the proof of the theorem formulated above is the construction of an algorithm transforming any natural number \(n\) into such a uniformly continuous operator \(\Phi_n\) from \(\overline{\mathcal E}_{2,2}\) into \(\overline{\mathcal E}_{2,2}\) that, whatever the point \(x\sigma y\) may be:
1) if \(x\sigma y\) belongs to \(-1\tau 1\), then \(\Phi_n\) is applicable to \(x\sigma y\) and \(\Phi_n(x\sigma y)\in -1\tau 1\).
2) if \(x\sigma y\) belongs to the boundary of the square \(-1\tau 1\), then
\[ \Phi_n(x\sigma y)=x\sigma y; \]
3) if \(x\sigma y\) does not belong to the lattice \(\mathfrak A(n)\,\square\, -1\Delta 1\), then \(\Phi_n(x\sigma y)\) belongs to the boundary of the square \(-1\tau 1\) and
\[ \Phi_{n+1}(x\sigma y)=\Phi_{n+2}(x\sigma y). \]
After the algorithms characterized above have been constructed, constructive operators \(G\) and \(F_0\) are constructed so that, for any point \(x\sigma y\),
\[ G(x\sigma y)\simeq \Phi_{\max(\mathfrak B(x),\,\mathfrak B(y))+1}(x\sigma y), \]
\[ F_0(x\sigma y)\simeq I(x\sigma y)-2^{-3}\cdot G(x\sigma y), \]
where \(I\) is the identity constructive operator from \(\overline{\mathcal E}_{2,2}\) into \(\overline{\mathcal E}_{2,2}\).
The continuity of the operators \(G\) and \(F_0\) follows from the main theorem of [3]. It is easy to see that the operator \(G\) is a retraction of the square \(-1\tau 1\) onto its boundary, while the operator \(F_0\) is a mapping of the square \(-1\tau 1\) into itself, without a fixed point, satisfying condition (1).
For the proof of Theorem 3, an algorithm is constructed which transforms any natural number \(n\) into such a uniformly continuous operator \(\varphi_n\) from \(\overline{\mathcal E}_{2,2}\) into the space of constructive real numbers that, for any point \(x\sigma y\) belonging to \(-1\tau 1\):
1) the operator \(\varphi_n\) is applicable to the point \(x\sigma y\), and \(0\leqslant \varphi_n(x\sigma y)\leqslant 1\);
2) \(\varphi_{n+1}(x\sigma y)=0 \equiv x\sigma y\in \mathfrak A(n)\,\square\, -1\Delta 1\).
It is easy to see that, for any point of the square \(-1\tau 1\), the identity algorithm \(h\) is a regulator of convergence into itself of the sequence \(g\)
such that, for every \(n\),
\[ g(n) \simeq \sum_{i=1}^{n} 2^{-i}\varphi_i(x\sigma y)\cdot \Phi_i(x\sigma y). \]
The limit of the sequence \(g\), constructed on the basis of the regulator of convergence in itself \(h\), will be denoted by \(W(x\sigma y)\). We now construct a constructive operator \(H\) such that, for any point \(x\sigma y\),
\[ H(x\sigma y) \simeq I(x\sigma y)-2^{-4}\cdot W(x\sigma y). \]
The operator \(H\) is a uniformly continuous mapping of the square \(-1\tau 1\) into itself without a fixed point.
We shall say that a mapping \(F\) of the square \(x_1\tau y_1\) into itself is a pseudouniformly continuous mapping if
\[ \forall n\,\neg\neg\exists m\,\forall xyzu\,((x\sigma y,z\sigma u\in x_1\tau y_1 \& \rho(x\sigma y \square z\sigma u)< \]
\[ <2^{-m}) \supset \rho(F(x\sigma y)\square F(z\sigma u))<2^{-n}). \]
This definition extends naturally to the case of an arbitrary constructive metric space.
The following theorem may be regarded as a constructive analogue of Brouwer’s fixed-point theorem.
Theorem 4. Whatever constructive pseudouniformly continuous mapping \(F\) of the square \(-1\tau 1\) into itself may be, for every \(l\) there is potentially realizable a rational point \(a\sigma b\), belonging to \(-1\tau 1\), such that
\[ \rho(F(a\sigma b)\square a\sigma b)<2^{-l}. \]
In the proof of this theorem one uses the fact that, for a piecewise-linear (simplicial) mapping of a square into itself, under which all vertices of both triangulations of the square are rational points, Brouwer’s theorem is transferred to constructive mathematics verbatim.
Let \(\overline{\mathcal E}_{3,2}\) denote the space of triples of constructive real numbers. We shall say that a point \(x\sigma y\sigma z\) of this space belongs to the unit two-dimensional sphere if \(x^2+y^2+z^2=1\). A constructive operator \(F\) from \(\overline{\mathcal E}_{3,2}\) into \(\overline{\mathcal E}_{3,2}\) will be called a vector field defined on the unit two-dimensional sphere if it is applicable to all points belonging to the unit two-dimensional sphere.
From Theorem 3 there follow the following theorems:
Theorem 5. One can construct a uniformly continuous tangent vector field, defined on the unit two-dimensional sphere and different from zero at all points belonging to the unit sphere.
Theorem 6. One can construct such a continuous tangent vector field \(F\), defined on the unit two-dimensional sphere, that for every point \(x\sigma y\sigma z\) belonging to the unit sphere,
\[ N(F(x\sigma y\sigma z))=1, \]
where \(N\) is the algorithm for computing the norm in the space \(\overline{\mathcal E}_{3,2}\).
Using the operator \(H\) from the proof of Theorem 3, construct a constructive uniformly continuous mapping \(H_1\) of the closed upper unit hemisphere into itself, without a fixed point, shifting all points
of the equator upward along the meridians. We transform the mapping \(H_1\) in the natural way into a constructive tangent uniformly continuous vector field \(J_1\) on the closed upper hemisphere. We now construct on the whole unit two-dimensional sphere such a vector field \(J\) that, whatever the point \(x\sigma y\sigma z\) may be:
1) if \(x\sigma y\sigma z\) belongs to the closed upper hemisphere, then
\[ J(x\sigma y\sigma z)=J_1(x\sigma y\sigma z); \]
2) if \(x\sigma y\sigma z\) belongs to the lower hemisphere, then
\[ J(x\sigma y\sigma z)=A\bigl(j_1(x\sigma y\sigma |z|)\bigr), \]
where \(A\) is the operator transforming a vector into the vector symmetric to it with respect to the vertical line passing through the initial point of the vector.
The operator \(J\) is a tangent uniformly continuous vector field defined on the unit two-dimensional sphere. At every point of the two-dimensional sphere it is different from zero. To prove Theorem 6, a constructive operator \(G_1\) is constructed such that for any point \(x\sigma y\sigma z\)
\[ G(x\sigma y\sigma z)\simeq \frac{1}{N(J(x\sigma y\sigma z))}\,J(x\sigma y\sigma z). \]
Remark 1. All constructive operators whose potential realizability is discussed in Theorems 1–6 can be made infinitely differentiable. The constructive operators \(G\), \(F_0\), and \(G_1\), constructed in the proofs of Theorems 1, 2, and 6, are locally uniformly continuous operators.
Remark 2. Any retraction of the square \(-1\tau 1\) onto its boundary and any normalized tangent vector field defined on the unit two-dimensional sphere do not satisfy the condition of pseudouniform continuity.
In conclusion, the author expresses gratitude to N. A. Shanin for his attention to the work.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
1 IV 1963
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