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
\( (\mathbb{T}^2 , F, bad measure) \).

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