## Inhomogeneous Markov shifts, their smooth realization and new examples of Anosov diffeomorphisms

#### Zemer Kosloff (University of Warwick)

Frank Adams Room 1, Alan Turing Building,

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 reasoning, that gives a construction of $$C^1$$ conservative
(satisfy Poincare’s reccurrence) Anosov Diffeomorphism of $$\mathbb{T}^2$$ without a Lebesgue
absolutely continuous invariant measure. By a theorem of Gurevic and Oseledec,
this can’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 a 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
$$H : \mathbb{T}^2 \to \mathbb{T}^2$$ such that $$H \circ F \circ H^{-1}$$ with Lebesgue measure is metric equivalent to
$$(\mathbb{T}^2 , F, bad measure)$$.