  Let $(X,T)$ and $(Y,S)$ be two subshifts so that $Y$ is a factor of $X$. For
any asymptotically sub-additive potential $\Phi$ on $X$ and $\ba=(a,b)\in\R^2$
with $a>0$, $b\geq 0$, we introduce the notions of $\ba$-weighted topological
pressure and $\ba$-weighted equilibrium state of $\Phi$. We setup the weighted
variational principle. In the case that $X, Y$ are full shifts with one-block
factor map, we prove the uniqueness and Gibbs property of $\ba$-weighted
equilibrium states for almost additive potentials having the bounded distortion
properties. Extensions are given to the higher dimensional weighted
thermodynamic formalism. As an application, we conduct the multifractal
analysis for a new type of level sets associated with Birkhoff averages, as
well as for weak Gibbs measures associated with asymptotically additive
potentials on self-affine symbolic spaces.
