SRB Measures and the Ledrappier-Young Dimension Formula

I present an exposition of Sinai-Ruelle-Bowen (SRB) measures and the Ledrappier-Young dimension formula, which relates the Hausdorff dimension of an invariant measure to its Lyapunov exponents and entropies. Beginning with measure disintegration along stable foliations (Rokhlin's theorem) and the formalism of conditional entropy, I examine how entropy contributions in different invariant directions influence the dimension of the measure. I then state and discuss the Ledrappier-Young formula in full detail, highlighting how entropy and expansion rates interact through a block decomposition of unstable foliations. I conclude with a synthesis of these ideas, emphasizing how the formula unifies dynamical invariants with fractal geometry of measures.