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Communications of the Moscow Mathematical Society

Densities of the Betti numbers of pre-level sets of quasi-periodic functions

A. I. Esterov For a smooth function f : M → R1 we denote by Mc the pre-level set {x ∈ M | f (x) ≤ c} (c ∈ R1 ). For almost every function f on a closed manifold M (that is, for any function except for a set of infinite codimension in the space of all functions) the topology of the set Mc (in particular, its Betti numbers) changes only when c passes through one of the finite number of critical values of f . For a periodic function f : Rk → R1 (having k linearly independent periods) the Betti numbers bj (Mc ) are infinite in general. It is not difficult to see that for almost every periodic function the limit bj (Bvk ∩ Mc ) βj (c) = lim (1) v→∞ V (Bvk ) exists, where Bvk ⊂ Rk is a closed ball of radius v with centre at the origin, and V (Bvk ) is the volume of this ball. The quantity βj (c) also varies only when c passes through one of the finite number of critical values of f . We now assume that f is a smooth quasi-periodic function. Definition 1. A quasi-periodic function is a composition f = φ ◦ P ◦ L of a linear monomorphism L : Rk → Rn , the natural factorization P : Rn → T n , and a smooth function φ : T n → R1 under the condition that the image P ◦ L is dense in T n . The function f generally has an infinite number of critical values, which are distributed in some way in R1 . In some models of chaos (see [1]) the question arises of the existence of the limit (1) and of the nature of its dependence on c. The existence of a limit of type (1) for the Euler characteristic of the set Mc and its piecewise-smoothness as a function of c were proved in [2]. Here we prove the following result. Theorem 1. For almost every quasi-periodic function f the limit (1) exists for any c and depends continuously on c. In the statement of the theorem “for almost every quasi-periodic function f ” means “for almost every function φ : T n → R1 defining f ”. Definition 2. A vanishing cycle of a critical point x0 of index α of a function f is a sphere S α−1 = {x ∈ M | u21 + · · · + u2α = δ; uα+1 = · · · = uk = 0} for sufficiently small δ, where u1 , . . . , uk are coordinates in a neighbourhood U of x0 for which f has the form f (x0 ) − u21 − · · · − u2α + u2α+1 + · · · + u2k . A vanishing cycle of a point x0 is not homologous to zero in U ∩ Mf (x0 ) \ {x0 }, but may be homologous to zero in the whole of Mf (x0 ) \ {x0 }. Definition 3. The radius of vanishing r(x0 ) of a critical point x0 is the minimal v for which the vanishing cycle of the critical point x0 is homologous to zero inside Bvk ∩ Mf (x0 ) \ {x0} (if no such v exists, then we set r(x0 ) equal to infinity). Now we can state two results from which Theorem 1 follows. Theorem 2. For almost every quasi-periodic function and for any c, j ∈ {0, . . . , k}, and r ∈ [0, ∞] there exists a finite density for the critical points in Mc of index j with radius of vanishing less than r. This density depends continuously on c. Then the same result is also true for the density for the critical points in Mc of index j with infinite radius of vanishing. AMS 2000 Mathematics Subject Classification. Primary 58K05; Secondary 26B35. DOI: 10.1070/RM2000v055n02ABEH000286

Communications of the Moscow Mathematical Society

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Theorem 3. The density βj (c) is equal to the difference between the density of the points in Mc of index j with finite radius of vanishing and the density of those of index j + 1 with infinite radius of vanishing (therefore it exists for almost every quasi-periodic function f and depends continuously on c). Proof of Theorem 2. This proof uses the following facts, contained in [2]. Let L : Rk → Rn be a linear monomorphism and let P : Rn → T n be the natural factorization, with the image of P ◦ L dense. Let M n−k be a submanifold of T n that is transversal to the image of P ◦ L. Then the quotient of the standard volume forms on Rn and Rk determines an orientation and a volume form ω on M n−k . Definition 4. The integral of the form ω over M n−k is called the transversal volume of M n−k and is denoted by Ve (M n−k ). (The transversal volume is always positive.) Theorem 4. Let K = (P ◦ L)−1(P ◦ L(Rk ) ∩ M n−k) be a discrete subset of Rk . If the transversal volume of M n−k is finite, then the density of the set K exists and is equal to this volume: #(K ∩ Bvk ) Ve (M n−k ) = lim , where # denotes the number of points in the set, Bvk ⊂ Rk is the v→∞ V (Bvk ) ball of radius v with centre at the origin, and V (B v ) is the volume of this ball. Let Mjn−k be the set of points x ∈ T n for which the restriction of df to the subspace L◦P (Rk ) ⊂ Tx T n ∼ = Rn is equal to zero, and the restriction of d2 f to this subspace is a non-degenerate quadratic form of index j. The existence of the densities mentioned in Theorem 2 follows from n−k defined by the an application of Theorem 4 to the set Mjn−k and to the open subsets Mj,r n−k ∩ Mc ) and Ve (M n−k ∩ conditions r(x) < r. These densities are respectively equal to Ve (M j,r

j

n−k Mc ) − Ve (Mj,∞ ∩ Mc ). Almost every φ : T n → R1 is not constant on any open subset of Mjn−k , and therefore the above volumes are continuous as functions of c for almost every quasi-periodic function f .

Proof of Theorem 3. We consider Nv = Bvk ∩ Mc (f ), where Bvk ⊂ Rk is the ball of radius v with centre at the origin. From standard Morse-theoretic arguments (applied to a manifold with boundary) it follows that the Betti number bj (Nv ) is equal to K1 −K2 +d, where K1 is the number of critical points x of index j whose vanishing cycles are homologous to zero in Nv \ {x}; K2 is the number of critical points x of index j + 1 whose vanishing cycles are not homologous to zero in Nv \ {x}; d is bounded in modulus by the total multiplicity of the degenerate critical points of the function f in Nv and of the critical points of the restriction f |∂Bv of f to the boundary of the ball. The quotient d/V (Bv ) tends to zero for almost every function f , since the total multiplicity of the degenerate points is finite for almost every function f , and the total multiplicity of the critical points of f |∂Bv grows proportionally to the (k − 1)-dimensional volume of the sphere ∂Bv . It follows from Theorem 2 that K1 /V (Bv ) and K2 /V (Bv ) tend to the density of critical points in Mc of index j with finite radius of vanishing and the density of those of index j + 1 with infinite radius of vanishing, respectively (when these densities exist). Passing to the limit as v → ∞ in the equality bj (Nv )/V (Bv ) = K1 /V (Bv ) − K2 /V (Bv ) + d/V (Bv ) gives the assertion of Theorem 3. Bibliography [1] G. M. Zaslavskii, M. Yu. Zakharov, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Pis’ma Zh. Eksper. Teoret. Fiz. 44 (1986), 349–353; English transl., JETP Lett. 44 (1986), 451–456. [2] S. M. Gusein-Zade, Funktsional. Anal. i Prilozhen. 23:2 (1989), 55–56; English transl., Functional Anal. Appl. 23 (1989), 129–130. Moscow State University Received 26/JAN/00