Let $\Phi$ be a homeomorphism of a compact metric space $M$ which preserves a regular Borel probability measure $\mu$.(`Regular' $\mu(U) > 0$, if U open. ) Under these hypothesis, I have two questions:

Q1. Is topologically transitivity of $\Phi$ equivalent to ergodicity of $\Phi$?

Q2. If $f$ is a continuous real valued function on $M$ is it true that the pointwise time averages $f^* (x) = lim_{N \to \infty} \Sigma_1 ^N f(\Phi ^j (x)) /N$ a la Birkhoff exist, not just for almost every x, but for every darn x?

Motivation. I just went through E. Hopf's proof (as presented in a 1971 BAMS article ) that for compact negatively curved surfaces the geodesic flow on the unit tangent bundle is ergodic. That proof gets much simpler if you get rid of the measure-theory/Birkhoff ergodic theorem business, which is to say, if the answer to flow versions of either Q1 or Q2 is `yes'.