Gregory Derfel

THE ZETA FUNCTION OF THE LAPLACIAN ON CERTAIN FRACTALS

We prove that the zeta function ζΔ of the Laplacian A on self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function, thereby answering a question posed by J. Kigami and M. Lapidus for this class of fractals.

On Bounded Solutions of the Balanced Generalized Pantograph Equation

The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in the early 1970s by T. Kato in connection with the analysis of the pantograph equation, y′(x)=ay(qx)+by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form −a 2 y″(x + a 1 y′(x) + y(x) = ∫ 0 ∞ μ (dα), where a 1 ≥ 0, a 2 ≥ 0 a 1 2 + a 2 2 > 0, and μ is a probability measure. By setting K:=∫ 0 ∞ ln ± μ(dα), we prove that if K≦0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K > 0 then such a solution exists. The result in the critical case, K=0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with “multiplication” jumps. The paper also includes three “elementary” proofs for the simple prototype equation y′(x)+y(x)=1/2y(qx)+1/2y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools.

Complex asymptotics of Poincaré functions and properties of Julia sets

The asymptotic behaviour of the solutions of Poincares functional equation f(?z) = p(f(z)) (? > 1) for p a real polynomial of degree = 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(z? F(log?z)), if f(z) ? 8 for z 8 and z W, where F denotes a periodic function of period 1 and ? = log? deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.

Functional-Differential and Functional Equations with Rescaling

A brief survey of the present state of functional- differential equations with rescaling is given. Various applications of equations with rescaling in probability, spectral theory of Schrodinger operator, subdivision processes and wavelets are discussed, as well.

On bounded continuous solutions of the archetypal equation with rescaling

The ‘archetypal’ equation with rescaling is given by y(x)=∬R2y(a(x−b))μ(da,db) (x∈R), where μ is a probability measure; equivalently, y(x)=E{y(α(x−β))}, with random α,β and E denoting expectation. Examples include (i) functional equation y(x)=∑ipiy(ai(x−bi)); (ii) functional–differential (‘pantograph’) equation y′(x)+y(x)=∑ipiy(ai(x−ci)) (pi>0, ∑ipi=1). Interpreting solutions y(x) as harmonic functions of the associated Markov chain (Xn), we obtain Liouville-type results asserting that any bounded continuous solution is constant. In particular, in the ‘critical’ case E{ln⁡|α|}=0 such a theorem holds subject to uniform continuity of y(x); the latter is guaranteed under mild regularity assumptions on β, satisfied e.g. for the pantograph equation (ii). For equation (i) with ai=qmi (mi∈Z, ∑ipimi=0), the result can be proved without the uniform continuity assumption. The proofs exploit the iterated equation y(x)=E{y(Xτ) | X0=x} (with a suitable stopping time τ) due to Doobs optional stopping theorem applied to the martingale y(Xn).

Divide-and-conquer recurrences — classification of asymptotics

Summary. We deal with a class of recurrence equations arising in the theory of algorithms, stochastic pocesses, computer science, etc. and give a classification theorem for asymptotics of their solutions. The method of the proof is based upon consideration of the exponential generating function, which is a solution of a linear nonhomogeneous functional equation with rescaling.

On the Asymptotic Behaviour of the Zeros of the Solutions of a Functional-differential Equation with Rescaling

We study the asymptotic behaviour of the solutions of a functionaldifferential equation with rescaling, the so-called pantograph equation. From this we derive asymptotic information about the zeros of these solutions.