Liouville theorems

Liouville's theorem on bounded entire analytic functions

If an entire function $ f (z) $ of the complex variable $ z = ( z _ {1}, \dots, z _ {n} ) $ is bounded, that is,

$$ | f (z) | \leq M < + \infty ,\ z \in \mathbf C ^ {n} , $$

then $ f (z) $ is a constant. This proposition, which is one of the fundamental results in the theory of analytic functions, was apparently first published in 1844 by A.L. Cauchy

for the case $ n = 1 $; J. Liouville presented it in his lectures in 1847, and this is how the name arose.

Liouville's theorem can be generalized in various directions. For example, if $ f (z) $ is an entire function in $ \mathbf C ^ {n} $ and

$$ | f (z) | \leq M ( 1 + | z | ^ {m} ) ,\ \ z \in \mathbf C ^ {n} , $$

for some integer $ m \geq 0 $, then $ f (z) $ is a polynomial in the variables $ ( z _ {1}, \dots, z _ {n} ) $ of degree not exceeding $ m $. Moreover, if $ u (x) $ is a real-valued harmonic function in the number space $ \mathbf R ^ {n} $, $ x = ( x _ {1}, \dots, x _ {n} ) $, and

$$ u (x) \leq M ( 1 + | x | ^ {m} ) \ ( \textrm{ or } - u (x) \leq M ( 1 + | x | ^ {m} ) ) , $$

$ x \in \mathbf R ^ {n} $, then $ u (x) $ is a harmonic polynomial in $ ( x _ {1}, \dots, x _ {n} ) $ of degree not exceeding $ m $ (see also ).

Liouville's theorem on conformal mapping

Every conformal mapping of a domain in a Euclidean space $ E ^ {n} $ with $ n \geq 3 $ can be represented as a finite number of compositions of very simple mappings of four kinds — translation, similarity, orthogonal transformation, and inversion. It was proved by J. Liouville in 1850 (see [2], Appendix 6).

This Liouville theorem shows the poverty of the class of conformal mappings in space, and from this point of view it is very important in the theory of analytic functions of several complex variables and in the theory of quasi-conformal mapping.

References

[1]	A.L. Cauchy, C.R. Acad. Sci. Paris , 19 (1844) pp. 1377–1384 Zbl 17.0200.02
[2]	G. Monge, "Application de l'analyse à la géométrie" , Bachelier (1850) pp. 609–616
[3]	A.V. Bitsadze, "Fundamentals of the theory of analytic functions of a complex variable" , Moscow (1972) (In Russian)
[4]	V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

E.D. Solomentsev

Liouville's theorem on approximation of algebraic numbers

A theorem stating that an algebraic irrationality cannot be very well approximated by rational numbers. Namely, if $\alpha$ is an algebraic number of degree $n \ge 2$ and $p$ and $q$ are any positive integral rational numbers, then $$ \left\vert{ \alpha - \frac{p}{q} }\right\vert \ge \frac{c}{q^n} $$ where $c$ is a positive constant depending only on $\alpha$ and expressible in explicit form in terms of quantities associated with $\alpha$.

By means of this theorem J. Liouville [1] was the first to construct non-algebraic (transcendental) numbers (cf. Transcendental number). Such a number is, for example, $$ \eta = \sum_{n} \frac{1}{2^{n!}} \,, $$ which is a series with rapidly-decreasing terms.

For $n=2$ Liouville's theorem gives the best possible result. For $n\ge3$ the theorem has often been strengthened. In 1909 A. Thue [2] established that for algebraic numbers $\alpha$ of degree $n\ge3$ and for $\nu > n/2+1$, \begin{equation}\label{eq:1} \left\vert{ \alpha - \frac{p}{q} }\right\vert \ge \frac{c}{q^\nu} \end{equation}

C.L. Siegel [3] improved Thue's result by showing that (1) is satisfied if $$ \nu > \min_{1\le s\le n-1} \left({ \frac{n}{s+1} + s }\right) $$ where $s$ is an integer, in particular, for $\nu > 2 \sqrt{n}$. Later F.J. Dyson [4] proved that \eqref{eq:1} holds when $\nu > \sqrt{2n}$. Finally, K.F. Roth [5] established that \eqref{eq:1} holds for any $\nu>2$. Roth's result is the best of its kind, since any irrational number $\xi$, algebraic or not, has infinitely many rational approximations $p/q$ satisfying the inequality \begin{equation}\label{eq:2} \left\vert{ \alpha - \frac{p}{q} }\right\vert < \frac{1}{q^2} \end{equation}

All strengthenings of Liouville's theorem mentioned above have one important deficiency — they are non-effective; namely: their methods of proof do not make it possible to establish how the constant $c$ in inequality \eqref{eq:1} depends on $\alpha$ and $\nu$. Effective strengthenings of Liouville's theorem have been obtained (see [6]–[8]), but only for values of $\nu$ that differ little from $n$.

References

[1]	J. Liouville, "Sur les classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques" C.R. Acad. Sci. Paris , 18 (1844) pp. 883–885; 910–911
[2]	A. Thue, "Ueber Annäherungswerte algebraischer Zahlen" J. Reine Angew. Math. , 135 (1909) pp. 284–305 Zbl 40.0265.01
[3]	C.L. Siegel, "Approximation algebraischer Zahlen" Math. Z. , 10 (1921) pp. 173–213 MR1544471 Zbl 48.0197.08 Zbl 48.0163.07
[4]	F.J. Dyson, "The approximation to algebraic numbers by rationals" Acta Math. , 79 (1947) pp. 225–240 MR0023854 Zbl 0030.02101
[5]	K.F. Roth, "Rational approximation to algebraic numbers" Mathematika , 2 (1955) pp. 1–20; 168 MR0077577 MR0072182
[6]	A. Baker, "Contributions to the theory of Diophantine equations I" Philos. Trans. Roy. Soc. London Ser. A , 263 (1968) pp. 173–191 MR0228424 Zbl 0157.09702
[7]	V.G. Sprindzhuk, "Rational approximations to algebraic numbers" Math. USSR Izv. , 5 (1971) pp. 1003–1019 Izv. Akad. Nauk SSSR Ser. Mat. , 35 (1971) pp. 991–1007 Zbl 0259.10032
[8]	N.I. Fel'dman, "An effective refinement of the exponent in Liouville's theorem" Math. USSR Izv. , 5 : 5 (1971) pp. 985–1002 Izv. Akad. Nauk. SSSR Ser. Mat. , 35 : 5 (1971) pp. 973–990

S.A. Stepanov

Comments

Rational approximations $p/q$ for which \eqref{eq:2} holds can be found among the convergents of the continued fraction expansion of $\xi$.

Liouville's theorem on the conservation of phase volume

The volume $ V $ of any domain $ G $ of the $ 6N $-dimensional phase space $ ( p , q ) $ (the space of components of the momenta $ p = ( \mathbf p _ {1}, \dots, \mathbf p _ {N} ) $ and coordinates $ q = ( \mathbf r _ {1}, \dots, \mathbf r _ {N} ) $ of each of the $ N $ particles of a classical system with potential forces of interaction) does not change in the course of time,

$$ V = \int\limits _ { (G) } d p d q = \textrm{ const } , $$

if all points of this domain are shifted in accordance with the equations of classical mechanics. The assertion is a consequence of the fact that the Jacobian of the transformation from the variables $ ( p , q ) $ (at time $ t $) to the variables $ ( p ^ \prime , q ^ \prime ) $ (at time $ t ^ \prime > t $) in accordance with the equations of motion (for example, in the form of Hamilton's equations) is equal to one. The quantity $ V $ is one of the integral invariants of Poincaré, and the theorem is a consequence of their existence. Liouville's theorem is used in statistical mechanics of classical systems (see Liouville equation). It was proposed by J. Liouville in 1851.

I.A. Kvasnikov