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 $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$.

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.

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.