Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a diffeomorphism, and let $\mu$ be a probability measure on $M$ with compact support.
As stated in the question Is there a name for a "stable" physical measure?, there are several different definitions of what it means for $\mu$ to be a "physical" measure. One possible defintion is as follows:
Definition. The measure $\mu$ is a physical measure of $f$ is there exists a neighbourhood $U$ of $\mathrm{supp}\,\mu$ such that for every probability measure $\nu$ with $\nu(U)=1$ that is absolutely continuous with respect to $\lambda$, we have $$ \frac{1}{n} \sum_{i=0}^{n-1} f^i_\ast\nu \ \overset{\text{weakly}}{\to} \ \mu \quad \text{as } n \to \infty.$$
A simple example is when $f$ has a stable fixed point or stable periodic orbit $A$, and $\mu$ is the equal-weight distribution on $A$.
However, I now want the convergence to hold without the Cesàro averaging; so, for example, this would exclude the case of a non-trivial periodic orbit.
Definition. The measure $\mu$ is a mixingly physical measure of $f$ is there exists a neighbourhood $U$ of $\mathrm{supp}\,\mu$ such that for every probability measure $\nu$ with $\nu(U)=1$ that is absolutely continuous with respect to $\lambda$, we have $$ f^n_\ast\nu \ \overset{\text{weakly}}{\to} \ \mu \quad \text{as } n \to \infty.$$
For example, in the notes of Marcelo Viana at http://w3.impa.br/~viana/out/sdds.pdf in Section 4 (starting at p79), the SRB measure on a uniformly hyperbolic attractor is a "mixingly physical measure". (Proposition 4.9 on p97 gives this fact with a description of the rate of the mixing behaviour.)
My question:
Is there a name for what I have called a "mixingly physical measure"? And/or is there any simple characterisation of mixingly physical measures? (E.g. "$\mu$ is mixingly physical if and only if $\mu$ is both mixing and physical", or "$\mu$ is mixingly physical if and only if $\mu$ is physical and $f$ is a mixing nonsingular transformation of $(M,\mathcal{B}(M),\lambda(\,\cdot\,\cap U))$"?)
