Thermodynamics of critical strange nonchaotic attractors.

The thermodynamic formalism is applied to dynamical attractors which have fractal geometry and on which all Lyapunov exponents are exactly zero. Such critical strange nonchaotic attractors (SNAs) which arise, for example, in the Harper system exhibit a static phase transition in the free energy. The Tsallis nonextensive entropy which is known to characterize the thermodynamics of systems with leading Lyapunov exponent zero is found to be subadditive for the critical states. These properties are shared by other quasiperiodic systems with critical SNAs.