Abundance of Observable Lyapunov Irregular Sets

Lyapunov exponent is widely used in natural science to find chaotic signal, but its existence seldom discussed. In the present paper, we consider problem of whether set points at which fails exist, called irregular set, has positive Lebesgue measure. The only known example with measure a figure-8 attractor by work Ott and Yorke (Phys Rev 78, 056203, 2008), whose key mechanism (homoclinic loop) easy be broken small perturbations. this show that surface diffeomorphisms robust homoclinic tangency given Colli Vargas (Ergod Theory Dyn Syst 21, 1657–1681, 2001), as well other several nonhyperbolic dynamics, have We can construct such sets both time averages exist do not on it.