UDC 513.821.838

MATHEMATICS

V. I. PUPKO, V. A. EFREMOVICH

THE BEHAVIOR OF EQUI-MORPHISMS AT INFINITY

(Presented by Academician P. S. Novikov on 20 X 1969)

In \((^{1})\) it was established that every equi-morphism \(f:\bar H^n \to H^{n'}\) of hyperbolic spaces preserves the tangency of curves with the absolute, as well as the tangency of curves lying on the absolute; however, the order of tangency with the absolute and the order of tangency of curves on the absolute, generally speaking, are not preserved. Here the properties of the equi-morphism \(f\) on the absolute will be clarified; namely, it turns out that although the order of tangency is not preserved, each of the three types of tangency—zero, finite, and infinite (see below)—remains invariant.

1. It is said that the tangency at a point \(a\) of two curves \(K\) and \(K_1\) in a metric space \(R\) has order \(\alpha\), \(0 \le \alpha \le \infty\), if for every \(\alpha' < \alpha\) one can find a common parametrization of the curves (a one-to-one and continuous correspondence between the parameters) under which
\[ xy/(ax)^{1+\alpha'} \to 0 \]
(\(x \in K\), \(y \in K_1\), \(x,y \to a\)), but for any \(\alpha' > \alpha\) no such parametrization exists\(^*\). In particular, for \(\alpha = 0\), \(0 < \alpha < \infty\), \(\alpha = \infty\), we obtain three types of tangency: zero, finite, and infinite.

Let us now consider a curve \(K_1\) lying on the absolute \(S^{n-1}\) of the space \(H^n\), and a curve \(K\), touching it at a point \(u \in S^{n-1}\), also lying on \(S^{n-1}\) or in \(H^n\). Here the order of tangency of the curves \(K\) and \(K_1\) is understood as the order of tangency of their images in the spherical Poincaré model (model P). If \(K_1\) is the central projection of \(K\) onto the absolute, then the order of tangency of \(K\) with \(K_1\) is called the order of tangency of \(K\) with the absolute.

Take a line leading to the point \(u \in S^{n-1}\) as the axis \(oz\) of a “cylindrical” coordinate system in \(H^n\): for any point \(p \in H^n\) denote by \(d\) its distance from \(oz\), and by \(z\) the projection of the vector \(\overrightarrow{op}\) onto \(oz\). Then the three introduced types of tangency of the curve \(K\) with the absolute can be distinguished, without resorting to a model, as follows.

Zero order of tangency: on \(K\) there exists a sequence of points \(p_n \to u\) for which \(d_n/z_n \to 0\) (\(d_n \to \infty\), \(p_n \to u\), \(p_n \in K\)); finite order: there exist two constants \(c, C > 0\) such that \(d/z > c\) for every point \(p \in K\), but there exists a sequence \(p_n \to u\) of points on \(K\) for which \(d_n/z_n < C\); infinite order: \(d/z \to \infty\) as \(p \to u\), \(p \in K\).

Consider, for example, a curve of zero order of tangency with the absolute. For it
\[ (1-r)/\theta \to 0,\quad \theta \to 0, \]
but
\[ (1-r)/\theta^{1+\alpha} \nrightarrow 0 \]
for any \(\alpha > 0\), where \(r=\operatorname{th}\rho/2\), \(\rho=op\), and \(\theta\) is the angle between \(op\) and \(oz\). The second relation means that
\[ 1/\operatorname{sh}\rho \cdot \theta^{1+\alpha} \nrightarrow 0 \]
and, consequently,
\[ \operatorname{sh}\alpha z/\operatorname{sh}d \nrightarrow 0 \]
as \(z \to \infty\), or
\[ d-\alpha z \nrightarrow \infty \]
as \(z \to \infty\). Hence it follows that for a sequence \(a_n \to 0\) there is on the curve a sequence of points \(p_n=(d_n,z_n)\) and a sequence \(c_n>0\) such that
\[ z_n \to \infty,\quad c_n/z_n \to 0,\quad d_n-a_n z_n < c_n, \]
i.e.
\[ d_n/z_n < a_n + c_n/z_n; \]
therefore \(d_n/z_n \to 0\). For finite (infinite) order of tangency
\[ \lim 1/\operatorname{sh}\rho \cdot \theta^{1+\alpha_0}=0 \]
for some \(\alpha_0>0\) (for every \(\alpha_0>0\)), i.e. for all positive \(\alpha \le \alpha_0\),
\[ d-\alpha z \to \infty, \]
whence, obviously, it follows that there is no sequence \(p_n \in K\) for which \(d_n/z_n \to 0\).

Remark. It is easy to see that if a curve \(K\) leading to the infinitely distant point \(u\) is subjected to a bounded shift, i.e. each of its points is moved a finite distance, then the order of its tangency with the absolute will not change.

\[ \rule{6em}{0.4pt} \]

\(^*\) For \(\alpha=0,\infty\) this formulation must be modified in a natural way.

The following theorems establish the invariance of each of the three indicated types of tangency.

2. Theorem 1. The zero order of tangency of a curve \(K\) with the absolute is invariant under equimorphisms of the space \(H^n\).

Proof. As has just been established, for some sequence \(p_n\) of points of the curve, \(d_n/z_n \to 0\) as \(p_n \to u\). We shall show, first, that \(d_n'/d_n\) remains bounded on both sides under any equimorphism \(f\). Denote by \(q_n\) the projection of the point \(p_n\) onto \(oz\), the images of the points \(p_n\) and \(q_n\) under the equimorphism \(f\) by \(p_n'\) and \(q_n'\), and the least distance of the point \(p_n'\) from the image of the axis \(oz\) by \(D_n'\). Then \(D_n' \leq p_n'q_n' \leq L\cdot p_nq_n = L\cdot d_n\) (for sufficiently large distances the equimorphism satisfies a two-sided Lipschitz condition; the constants entering here will be denoted by \(l\) and \(L\)). Further, \(d_n' \leq D_n' + c_0\), where \(c_0\) is the greatest deviation of the image of \(oz\) from \(oz\) (see (2)); and we assume that the points \(u\) and \(o\) remain fixed. Thus
\[ d_n' \leq L\cdot d_n + c_0 < (L+1)d_n=\tilde L\cdot d_n . \]
Considering the inverse mapping \(f^{-1}\), in the same way we verify that \(d_n' > \tilde l\cdot d_n\). Hence
\[ \tilde l < d_n'/d_n < \tilde L, \]
where \(\tilde l,\tilde L>0\) are certain constants. We now finish the proof of the invariance of the zero order of tangency. Similarly to the preceding, we have
\[ l<\rho_n'/\rho_n<L, \]
where \(\rho_n=op_n,\ \rho_n'=op_n'\). Hence
\[ l/2<(z_n'+d_n')/(z_n+d_n)<2L, \]
whence it follows that
\[ \frac l2<\frac{z_n'/z_n+d_n'/z_n}{1+d_n/z_n}<2L, \]
and since \(d_n/z_n\to0\) and \(d_n'/z_n\to0\) (the latter by virtue of the boundedness of the ratio \(d_n'/d_n\)) as \(n\to\infty\), it follows that
\[ l/3<z_n'/z_n<3L. \]
Therefore
\[ \frac{d_n'}{z_n'}=\frac{d_n'}{d_n}\frac{d_n}{z_n}\frac{z_n}{z_n'}\to0, \]
as was required.

Before formulating Theorem 2, let us prove one auxiliary proposition, which will also be needed later.

Lemma. In order that two curves \(K_1,K_2\subset S^{n-1}\) have order of tangency at the point \(u\): a) zero, b) infinite, it is necessary and sufficient that on each projecting cone \(oK_1\) and \(oK_2\) there be found two curves \(\mathcal K_1\) and \(\mathcal K_2\), deviating boundedly from one another, such that: a) \(\mathcal K_1\) and \(\mathcal K_2\) touch the absolute; any two such curves have order of tangency with the absolute not exceeding zero; b) \(\mathcal K_1\) and \(\mathcal K_2\) have infinite order of tangency with the absolute.

Proof. Case a). Necessity. Let the order of tangency of \(K_1\) with \(K_2\) be equal to zero. Let \(u_1\in K_1,\ u_2\in K_2\). If by \(\beta\) we denote the angle \(u_1ou_2\), and by \(\psi_1\) and \(\psi_2\) the angles \(u_1ou\) and \(u_2ou\), respectively, then
\[ \lim \beta/\psi_i=0;\qquad \beta/\psi_i^{1+\alpha}\nrightarrow 0,\quad i=1,2;\ \alpha>0. \]
On the projecting cone \(oK_1\) construct a curve \(\mathcal K_1\) with equation \(1-r_1=\beta\). Then
\[ (1-r_1)/\psi_1=\beta/\psi_1\to0 \]
and
\[ (1-r_1)/\psi_1^{1+\alpha}\nrightarrow0, \]
which means a zero order of tangency of \(K_1\) with the absolute. In exactly the same way, on \(oK_2\) construct a curve \(\mathcal K_2\) with equation \(1-r_2=\beta\) (i.e., put \(r_1=r_2=r=1-\beta\), where \(r=\operatorname{th}\rho/2\)). Denote the distance between corresponding points on \(\mathcal K_1\) and \(\mathcal K_2\) by \(d\). Then
\[ \operatorname{sh}\frac d2=\operatorname{sh}\rho\,\sin\frac\beta2 =\frac{2r}{1+r}\frac{\sin\beta/2}{1-r}\to\frac12 \]
as \(u_{1,2}\to u\).

Now suppose that there are \(\overline{\mathcal K}_1\subset oK_1\) and \(\overline{\mathcal K}_2\subset oK_2\), boundedly deviating from each other, and that at least one of them has order of tangency with the absolute greater than zero. Then, by the remark to item 1, it follows that the second curve will have the same order; denote it by \(\alpha_0\). Let the boundedness of the distance between \(\overline{\mathcal K}_1\) and \(\overline{\mathcal K}_2\) be attained under the correspondence \(x_1\leftrightarrow x_2\) \((x_1\in\overline{\mathcal K}_1,\ x_2\in\overline{\mathcal K}_2)\). From the triangle \(ox_1x_2\) we have
\[ \beta^2\leq c(1-r_1)(1-r_2),\qquad c=\text{const}. \]
Consider the ratio \(\beta/\psi_1^{1+\alpha}\), where \(0<\alpha<\alpha_0\). Then
\[ \frac{\beta^2}{\psi_1^{(1+\alpha)2}} \leq \frac{c(1-r_1)(1-r_2)}{\psi_1^{2(1+\alpha)}} = c\,\frac{1-r_1}{\psi_1^{1+\alpha}}\frac{1-r_2}{\psi_1^{1+\alpha}}. \]
Since \(\psi_1\) and \(\psi_2\) are equivalent infinitesimals (see (1)), both of the last factors tend to zero. Therefore \(\beta/\psi_1^{1+\alpha}\to0\), and hence the order of tangency of \(K_1\) with \(K_2\) is not less than \(\alpha_0\).

Sufficiency. Instead of the required assertion, we shall give the formulation of an assertion equivalent to it. If \(K_1\) and \(K_2\) have at the point \(u\in S^{n-1}\) contact of finite (or more than finite) order, then on the projecting cones \(oK_1\) and \(oK_2\) one can find curves \(\mathcal K_1\) and \(\mathcal K_2\) having finite order of contact with the absolute and deviating from one another in a bounded manner. The proof of this fact is entirely analogous to the preceding proof.

Case b). Necessity. Just as in the preceding case, we construct curves \(\mathcal K_i,\ 1-r_i=\beta,\ i=1,2\), which, obviously, have infinite order of contact with the absolute.

Sufficiency. Suppose that on \(oK_1\) and \(oK_2\) there are curves \(\mathcal K_1\) and \(\mathcal K_2\) at a bounded distance from one another, the order of contact of each of them with the absolute being infinite. Taking, as above, \(x_1\leftrightarrow x_2\) as the correspondence between these curves and considering the triangle \(ox_1x_2\), we obtain, as before,
\(\beta^2\leq c(1-r_1)(1-r_2)\), where \((1-r_1)/\psi_1^N\to 0\) and \((1-r_2)/\psi_2^N\to 0\) for any \(N\) greater than or equal to one. In this case
\(\beta^2/\psi_1^{2N}\leq c(1-r_1)(1-r_2)/\psi_1^{2N}\to 0\), as was required.

Theorem 2. The zero order of contact of curves \(K_1\) and \(K_2\) lying on the absolute is invariant with respect to equimorphisms of the space \(H^n\).

Proof. Denote the images of the curves \(K_1\) and \(K_2\) under the equimorphism \(f\) by \(K_1'\) and \(K_2'\). Suppose that the order of contact of \(K_1'\) with \(K_2'\) is finite (or infinite). Then on the projecting cones \(o'K_1'\) and \(o'K_2'\) there will be found curves \(\mathcal K_1'\) and \(\mathcal K_2'\), deviating from one another in a bounded manner and having finite order of contact with the absolute. In this case the equimorphism \(f^{-1}\) takes \(\mathcal K_1'\) and \(\mathcal K_2'\) into curves \(\widetilde{\mathcal K}_1\) and \(\widetilde{\mathcal K}_2\) with finite (nonzero) order of contact with the absolute. Projecting orthogonally the curves \(\widetilde{\mathcal K}_1\) and \(\widetilde{\mathcal K}_2\) onto the cones \(oK_1\) and \(oK_2\), respectively, we obtain curves \(\mathcal K_1\) and \(\mathcal K_2\), lying at a bounded distance from \(\widetilde{\mathcal K}_1\) and \(\widetilde{\mathcal K}_2\) and, consequently, having at least finite order of contact with the absolute. If the distance between \(\mathcal K_1\) and \(\mathcal K_2\) is bounded, then the curves \(K_1\) and \(K_2\) could not have contact of zero order; if, however, it is unbounded, then \(\mathcal K_1'\) and \(\mathcal K_2'\) also could not lie at a finite distance from one another. The theorem is proved.

3. Theorem 3. The infinite order of contact with the absolute of a curve \(K\) leading to an infinitely remote point \(u\) is an invariant of equimorphisms of the space \(H^n\).

The proof rests on somewhat different considerations than the proof of Theorem 1.

Theorem 4. The infinite order of contact of curves \(K_1\) and \(K_2\) lying on the absolute is invariant with respect to equimorphisms of \(H^n\).

Proof. Denote the images of the curves \(K_1\) and \(K_2\) by \(K_1'\) and \(K_2'\). By virtue of the lemma of paragraph 1, on the projecting cones \(oK_1\) and \(oK_2\) there exist curves \(\mathcal K_1\) and \(\mathcal K_2\), deviating from one another in a bounded manner and having infinite order of contact with the absolute. This means that the projections of their images \(\mathcal K_1'\) and \(\mathcal K_2'\) onto the projecting cones \(o'K_1'\) and \(o'K_2'\) deviate from one another in a bounded manner and also touch the absolute with infinite order of contact. From the already mentioned lemma it follows that the order of contact of the curves \(K_1'\) and \(K_2'\) is infinite, as was required.

From the preceding theorems follows the validity of the following theorem.

Theorem 5. The finite order of contact of a curve with the absolute, as well as the finite order of contact of two curves lying on the absolute, remains finite for their images under equimorphisms of the space, i.e., finiteness of the order of contact is invariant with respect to equimorphisms of the space \(H^n\).

Steklov Mathematical Institute
Academy of Sciences of the USSR
Moscow

Received
16 X 1969

CITED LITERATURE

1. V. A. Efremovich, V. I. Pupko, DAN, 160, No. 1 (1965).
2. V. A. Efremovich, E. S. Tikhomirova, DAN, 152, No. 5 (1963).