A volume preserving nonuniformly hyperbolic diffeomorphism with arbitrary number of ergodic components and close to the identity

We prove that for any $\mathrm{\ell }\in \text{NN}\cup \{\mathrm{\infty }\}$ and any $r\in \text{NN}$, every compact smooth Riemannian manifold $\text{cM}$ of $\dim \text{cM}\ge 5$ carries a $C^{\mathrm{\infty }}$ volume preserving nonuniformly hyperbolic diffeomorphism, which has exactly $\mathrm{\ell }$ ergodic components (in fact, Bernoulli components) and is $C^r$ close to the identity.