In this paper, we study the Hausdorff dimension of generalized intrinsic level set with respect to given ergodic meausre in a class non-uniformly hyperbolic interval maps finitely many branches.

Given an orbit whose linearization has invariant subspaces satisfying some non-resonance conditions in the exponential rates of growth, we prove existence of invariant manifolds tangent to these subspaces. The exponential rates of growth can be understood either in the sense of Lyapunov exponents or in the sense of exponential dichotomies. These manifolds can correspond to “slow manifolds”, whi...