This question is a follow up to: this post.

To let $ X $ be a separable Banach room, $ phi in C (X; X) $ be an injectively continuous non-affine card, and $ A $ be tight $ G _ { delta} $ Subset of $ X $, How big can $ orb ( phi, A) $ Be? Where:

$$

begin {align}

& orb (A) triangleq bigcup_ {a in A} orb (A), \

& orb ( phi, a) triangleq { phi ^ n (a) } _ {n in mathbb {N}},

end

$$

What can we say about the size of …? $ orb ( phi, A) $? More precisely, does $ X $ satisfy either:

- If $ mu $ is a non-atomic and strictly positive Borel measure $ X $, then $ orb ( phi, A) $ from full $ mu $-measure up,
- is $ orb ( phi, A) $ Hair-zero,
- $ orb ( phi, A) $ covers $ X-E $, from where $ E $ is a finite dimensional subspace of $ X $?