Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates

Let $$M$$ a two-dimensional compact manifold and $$f:M\to M$$ a diffeomorphism $$C^r$$, $$r\geq 2$$ and $$f(x,y)=(mx,\lambda y)$$ where $$m:M\to \mathbb{R}$$ and $$\lambda:M\to \mathbb{R}$$ ,$$\lambda<1.

That is clear when $$m$$ and $$\lambda$$ are constant, $$\Lambda\subset M$$ is a horseshoe for $$f$$. I am looking for situation when $$\lambda$$ and $$m$$ are not constant then we have horseshoes.There is a sentence, they said "we can replace by supremum over all invariant Borel probability measures of ergodic averages of ratios of rates".

Could one explain me what does it mean "supremum over all invariant Borel probability measures of ergodic averages of ratios of rates"? I mean to write mathematical notions.