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For a $C^1$ non-conformal repeller, this paper proves that there exists an ergodic measure of full Carathéodory singular dimension. For an average conformal hyperbolic set of a $C^1$ diffeomorphism, this paper constructs a Borel probability measure (with support strictly inside the repeller) of full Hausdorff dimension. If the average conformal hyperbolic set is of a $C^{1+\alpha }$ diffeomorphism, this paper shows that there exists an ergodic measure of maximal dimension.

This paper considers C 2 random dynamical systems on a Banach space, and proves that under some mild conditions, SRB measures are characterized by invariant measures satisfying the Pesin entropy formula, in which entropy is equal to the sum of positive Lyapunov exponents of the system. This result is a random version of the main result in A. Blumenthal and L.-S. Young's paper [12].

Let $f:X\to X$ be an invertible Lipschitz transformation on a compact metric space $X$. Given a H"{o}lder continuous invertible operator cocycles on a Banach space and an $f$-invariant ergodic measure, this paper establishes the H"{o}lder continuity of Oseledets subspaces over a compact set of arbitrarily large measure. This extends a result in \cite{Simion16} for invertible operator cocycles on a Banach space. This paper also proves the H"{o}lder continuity in the non-invertible case. Finally, some applications are given in the end of this paper.

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