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We study the fiber Lyapunov exponents of step skew-product maps over acomplete shift of $N$, $N\ge2$, symbols and with $C^1$ diffeomorphisms of thecircle as fiber maps. The systems we study are transitive and genuinelynonhyperbolic, exhibiting simultaneously ergodic measures with positive,negative, and zero exponents. Examples of such systems arise from theprojective action of $2\times 2$ matrix cocycles and our results apply to anopen and dense subset of elliptic $\mathrm{SL}(2,\bR)$ cocycles. We derive amultifractal analysis for the topological entropy of the level sets of Lyapunovexponent. The results are formulated in terms of Legendre-Fenchel transforms ofrestricted variational pressures, considering hyperbolic ergodic measures only,as well as in terms of restricted variational principles of entropies ofergodic measures with a given exponent. We show that the entropy of the levelsets is a continuous function of the Lyapunov exponent. The level set of thezero exponent has positive, but not maximal, topological entropy. Under theadditional assumption of proximality, as for example for skew-products arisingfrom certain matrix cocycles, there exist two unique ergodic measures ofmaximal entropy, one with negative and one with positive fiber Lyapunovexponent.

Entropy spectrum of Lyapunov exponents for nonhyperbolic step skew-products and elliptic cocycles