A few explicit examples of complex dynamics of inertia groups on surfaces - a question of Professor Igor Dolgachev

I give a few explicit examples which answer an open minded question of Professor Igor Dolgachev on complex dynamics of the inertia group of a smooth rational curve on a projective K3 surface 

"It is challenging to find (further) pairs $(S, C)$ of a projective K3 surface and a smooth rational curve $C \subset S$ such that ${\rm Ine}\,(C)$ contains an element with positive topological entropy" and its variants for a rational surface and a non-projective K3 surface.