The Box Dimension For Invariant Graphs Of Skew Product Systems In Dimension Three

The estimation on the dimensions of hyperbolic sets is an important area in dif-ferentiable dynamical systems which has been widely and continuously concerned by famous scholars at home and abroad.In this paper,we consider the Box dimension for invariant graphs of skew product systems in dimension three.Specifically,let f be a diffeomorphism on some compact surface and let(?)be a basic set of f.Let F be a C~2 diffeomorphism on (?)×R,with the following structure:F:(?)×R?(?)×R,F(x,t)=(f(x),g_x(t)),where f is uniformly hyperbolic and g:M?Diff~2(R)is uniformly expanding which is weaker than f.Under this assumption,it is well known by the classical theory that there exists a continuous invariant graph?.We focus on the Box dimension of?re-stricted on the unstable manifold(with respect to F).By improving several works of Hu,Walkden and Díaz-Gelfert-Gr(?)ger-J(?)ger et al.,we show that the Box dimension of ? is ?^c+?^u Where ?^c is some transversal Box dimension in the direction of R and ?^u is the solution to some Bowen's pressure equation.Our result improves the work of Walkden in the three-dimensional case and extends several results of Díaz-Gelfert-Gr(?)ger-J(?)ger.The main difference from previous works is that our estimation of the Box dimension is indeed an equality which provides more precise information.In ad-dition,by using the pressure equation,our argument does not depend on the invariant graph being Lipschitz(or not)which provides a more general and innovative result.