# Probability – size of the orbit of a dense crowd

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 $$X$$?