# zbMATH — the first resource for mathematics

Singular perturbation of symbolic flows and poles of the zeta functions. (English) Zbl 0708.58019
This work is motivated by studies of scattering by an obstacle $${\mathcal O}$$ which consists of several strictly convex bodies. In this context, a function $$F_ D(s)$$ can be defined by means of the geometry of periodic rays in the exterior of $${\mathcal O}$$. A theorem of Ikawa says that if $$F_ D(s)$$ cannot be analytically continued to an entire function, then the scattering matrix S(z) for $${\mathcal O}$$ has an infinite number of poles in the half-plane $$\{$$ z: $$Im(z)<\alpha \}$$ for some $$\alpha >0$$. This, in turn, means that the modified Lax-Phillips conjecture is valid for $${\mathcal O}.$$
In general, it is difficult to show that an analytic continuation of $$F_ D(s)$$ to an entire function is impossible. However, $$F_ D(s)$$ is closely related to the zeta function $$\zeta$$ (s) of a certain associated symbolic flow; specifically, the singularities of -(d/ds)log $$\zeta$$ (s) in a specified domain coincide with those of $$F_ D(s)$$. Thus, consideration of the singularities of $$F_ D(s)$$ is essentially reduced to analysis of $$\zeta$$ (s). Unfortunately, it is also difficult to find the singularities of -(d/ds)log $$\zeta$$ (s).
The author shows that, if the convex bodies are small compared to the distances between them, then it is possible to find a pole of $$\zeta$$ (s) corresponding to $${\mathcal O}$$. The author’s main theorem is used to show that the modified Lax-Phillips conjecture is valid for obstacles of this type.
Reviewer: W.J.Satzer jun

##### MSC:
 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 37C10 Dynamics induced by flows and semiflows 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 35P25 Scattering theory for PDEs