Multifractal Analysis For Birkhoff Averages On The Skew-product Case

In this paper, we mainly consider the multifractal analysis for Birkhof averages ina given system. For example,we can see [9] at page157, given a mapping g:[0,1]~2â†'[0,1]~2, g(x, y)=(lx, my)(l≥m≥2), then we know g can contract the unit square intoseveral small rectangles, and their length and width are lm. If iterating f time andtime again, at last we will get an infinite number of limit points. All these points formthe limit set Λ. Accordingly, we can define the set Λ α(seeing the following section), thenconsider dimHΛ α.While in this text, we consider the general mapping. Given>0small, we consideran C~2perturbation f of g, of the form f(x, y)=(a(x, y), b(y)). If iterating f onetime, we will get several distorted rectangles. We can know from [2] that the length ofthese distorted rectangles can be expressed by the form of the derivative of a(x, y), b(y).Then we also can get the limit set Λ similarly, and we can define the set Λ_α~ψ, and con-sider dim_H Λ_α~ψ. In the following section, we will study the relationship of the Hausdorfdimension!Kolmogorov-Sinai entropy and Lyapunov exponents from the perspective ofsymbolic dynamical system on this skew-product case.