Almost periodic mappings to complex manifolds. S. Favorov, N. Parfyonova 7 Abstract. H.Bohr in 1930 proved that if a holomorphic bounded function on a strip is 0 almost periodic on a straight line in the strip, then it is almost periodic on the whole strip. 0 We find some conditions when the result is valid for holomorphic mappings of tube domains to 2 various complex manifolds. n a J A continuous mapping F of a tube T = z = x+iy : x Rm,y K Rm K 9 { ∈ ∈ ⊂ } 2 to a metric space X is almost periodic if the family F(z+t) t Rm of shifts along Rm is a relatively compact set with respect to t{he topol}og∈y of the uniform ] V convergence on T . C K Further, let X be a complex manifold, and F be a holomorphic mapping of . h at a tube TΩ = z = x+iy : x Rm,y Ω , with the convex open base Ω Rm, { ∈ ∈ } ⊂ m to X. We will say that F is almost periodic if the restriction of F to each tube [ T , with the compact base K Ω, is almost periodic. K 1 ⊂ For X = C we obtain the well-known class of holomorphic almost periodic v 0 functions; for X = Cm the corresponding class was being studied in [3], [7], 6 8 [8], [6]; for X = CP, we get the class of meromorphic almost periodic functions 1 0 that was being studied in [4], [10], [5]; the class of holomorphic almost periodic 7 0 curves, corresponding to the case X = CPm, was being studied in [6]. / h The following theorem is well known: t a Theorem B (H. Bohr [2]) If a holomorphic bounded function on a strip is m almost periodic on some straight line in this strip, then this function is almost : v i periodic on the whole strip. X r This theorem was extended to holomorphic functions on a tube domain in a [11]; besides usual uniform metric, various integral metrics were being studied there. The direct generalization of Theorem B to complex manifold is not true: Example. Let f(z) be the quotient of two periodic functions sin√2z and sinz. It is clear that the restriction of f(z) to any straight line z = x + iy : 0 { x R ,y = 0, is almost periodic on this line. Besides, f(z) is bounded as a 0 ∈ } 6 mapping to the compact manifold CP. Nevertheless, zeros and poles of f(z) are not separated on the real axis z = x R, therefore f(z) is not almost periodic ∈ on any substrip containing the real axis ( see [4]). In order to give the right version of Theorem B, we need the following propo- sition. Proposition. If K is compact subset of Rm, then an almost periodic map- ping F of T to a metric space X is uniformly continuous and F(T ) is a K K relatively compact subset of X. Proof. Let ϕ(t) = supd(F(z + t),F(z)), t Rm. ∈ y K ∈ It is easy to prove that ϕ(t) is an almost periodic function on Rm, therefore the function ϕ(t) extends continuously to Bohr’s compactification B of the set Rm (see, for example, [1]). Hence for any point τ B and any ε > 0 there exists a ∈ neighborhood U B of this point such that ⊂ ϕ(t) ϕ(t ) < ε ′ | − | for each t,t U Rm. ′ ∈ ∩ Using the obvious inequality ϕ(t t ) ϕ(t) ϕ(t ) ′ ′ − ≤ | − | we get for all z T ,t,t U Rm K ′ ∈ ∈ ∩ d(F(z + t ),F(z + t)) ε. ′ ≤ Therefore the function F(τ,y) = lim F(x + iy) x τ, x Rm → ∈ is well-defined and continuously maps the compact set B K to X. Since × F(x,y) = F(x+iy) for x Rm, we obtain all the statements of our proposition. ∈ Note that a bounded holomorphic function on a tube domain is uniformly continuous on every subtube with the compact base, but bounded holomorphic mappings, in general, have no this property. Therefore the following result is natural. Theorem. Let F be a holomorphic mapping of T to a complex man- Ω ifold X such that for every compact subset K Ω the mapping F is uniformly ⊂ continuous on T and F(T ) is a relatively compact subset of X. If the restric- K K tion of F(z) to some hyperplane Rm + iy is almost periodic, then F(z) is an ′ almost periodic mapping of T to X. Ω Corollary. Let F be a holomorphic mapping form T to a compact complex Ω manifold X such that F is uniformly continuous on T for every compact set K K Ω. If the restriction of F(z) to some hyperplane Rm+iy is almost periodic, ′ ⊂ then F(z) is an almost periodic mapping of T to X. Proof the Theorem. Ω Take an arbitrary sequence t Rm. Since the function F(z) is uniformly n { } ⊂ continuous, the family F(z+t is equicontinuous on each compact set S T . n Ω { } ⊂ Further, it follows from the condition of the Theorem that the union of all the images of S under mappings of this family is contained in a compact subset of X. Therefore, passing on to a subsequence if necessary, we may assume that the sequence F(z + t ) converges to a holomorphic mapping G(z) uniformly on n { } every compact subset of T . It easy to see that the mapping G(z) is bounded Ω and uniformly continuous on every tube T with the compact base K Ω. Let K ⊂ us prove that this convergence is uniform on every T . Assume the contrary. K Then we get d(F(z + t ),G(z )) ε > 0 (1) n n n 0 ≥ for some sequence z = x + iy T , where K is some compact subset n n n K′ ′ ∈ of Ω. Replacing sequence by a subsequence if necessary, we may assume that the mappings G(x + z) converge to a holomorphic mapping H(z), and the n ˜ mappings F(z + x + t ) converge to a holomorphic mapping H(z) uniformly n n on every compact subsets of T . We may also assume that y y K . Using Ω n 0 ′ → ∈ (1) we get ˜ H(iy ) H(iy )) ε . 0 0 0 | − | ≥ Since the mapping F(x+iy ) of Rm to X is almost periodic, we may assume that ′ a subsequence of the mappings F(x+t +iy ) converges to G(x+iy ) uniformly n ′ ′ in x Rm. Therefore the sequences of mappings F(x + x + t + iy ) and n n ′ ∈ G(x+x +iy ) have the same limit, i. e., H˜(x+iy ) = H(x+iy ) for all x Rm. n ′ ′ ′ ∈ ˜ ˜ Since H(z), H(z) are holomorphic mappings, we get H(x+iy) H(x+iy) on ≡ T . This contradiction proves the Theorem. Ω REFERENCES [1] Berg Ch. Introduction to the almost periodic functions of Bohr // Proceed- ings of a symposium held in Copenhagen, April 24-25. - 1987. - P.15-24. [2] Bohr H. Ueber analytische fastperiodische Funktionen // Math. Ann. - 1930 - Vol. 103 -P. 1-14. [3] Favorov S.Ju., Rashkovskii A.Ju., Ronkin L.I. Almost periodic divisors in a strip // Journal D’analyse Mathematique. - 1998. - Vol.74 - P.325-345. [4] Parfyonova N.D., Favorov S.Yu. Meromorphic almost periodic functions // Math. Stud. Lviv. - 2000. - Vol. 13. - P.190-198. [5] Parfyonova N.D. Meromorphic almost periodic functions and their continu- ous extension on the Bohr’s compact // Vestnik KharkovskogoUniversiteta - 2002. - Vol. 542. - P. 73-84. 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