Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure
Markov partitions introduced by Sinai and Adler and Weiss are a tool that enables transfering questions about ergodic theory of Anosov Diffeomorphisms into questions about Topological Markov Shifts and Markov Chains. This talk will be about a reverse reasonning, that gives a construction of $C^{1}$ conservative (satisfy Poincaretextquoteright s reccurrence) Anosov Diffeomorphism of $mathbb{T}^{2}$ without a Lebesgue absolutely continuous invariant measure. By a theorem of Gurevic and Oseledec, this cantextquoteright t happen if the map is $C^{1+alpha}$ with $alpha>0$. Our method relies on first choosing a nice Toral Automorphism with a nice Markov partition and then constructing bad conservative Markov measure on the symbolic space given by the Markov partition. We then push this measure back to the Torus to obtain a bad measure for the Toral automorphism. The final stage is to find by smooth realization a conjugating map $H$ such that $Hcirc fcirc H^{-1}$ with Lebesgue measure is metric equivalent to $left(mathbb{T}^{2},mathcal{B},f, Bad measureright)$.