Dimension Estimate of Almost Periodic Attractors by Simultaneous Diophantine Approximation

In this paper we estimate fractal dimensions of almost periodic trajectories for a semilinear parabolic partial differential equation: ∂u/∂t−d(t) Δu+g(t) u f(t), where we assume periodicity:d(t+α1)=d(t),g(t+α2)=g(t) for irrational numbersα1, α2, which are linearly independent over the rationals, andf(t+1)equals;f(t). By using simultaneous Diophantine approximation, we can show that the dimension of the almost periodic attractor is majorized by 1/γ1+2/γ2, whereγ1is the minimum number of the exponents of Hölderʹs conditions on the periodic functions andγ2is the secondary minimum.