I would like to better understand this sentence taken from Page 281 of Topological Methods in Hydrodynamics By Vladimir I. Arnold, Boris A. Khesin
Even for the $L_1$-norm, one can provide such growth of the $E_1$-magnetic energy if the number of connected components of the intersection $R_t \cap R$ increases exponentially with time $t$.
Being $R_t=g^t R$, on what $R$ applied to a diffeomorphism $g^t=e^{\lambda t}$.
Furthermore, the magnetic energy of a field $B$ with respect to the $L_d$ norm is given by$$E(B)_{L_d(M)}= \int_M B^d \mu$$ (where $M$ is a manifold and $\mu$ is a volume form).
My question is: Because if the number of connected components of $R_t \cap R$ grows exponentially then the magnetic energy of field B grows exponentially with respect to the $L_1$-norm?
Below is a picture from the book on Page 280 which shows that the magnetic energy of the field grows exponentially to the $L_2$ norm.
