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Entropy and twisted cohomology. (English) Zbl 0611.58036
This substantial paper is concerned with variations of Shub’s entropy conjecture which (in its first version) says: The topological entropy h(g) of a smooth map $$g: M\to M$$ on a compact simply connected manifold M is bounded from below by the spectral radius of the linear map induced by g on cohomology. In the non simply connected case passing to a finite cover can produce larger cohomology groups and hence may increase this spectral radius - the above formulation of the conjecture is not adequate here. A proper version of the entropy conjecture for the general (non simply connected) case uses twisted cohomology and reads $$h(g)\geq \gamma_ i(g,\rho)$$, where $$\rho$$ is any representation of the fundamental group $$\pi_ 1M$$ by linear isometries of some seminormed vector space, $$\gamma_ i(g,\rho)$$ is the i-dimensional growth rate for $$\rho$$-twisted cohomology. For each i an ’optimal’ $$\rho_ i$$ is obtained. For special cases the twisted cohomology entropy conjecture is proved following the existing arguments for the untwisted one - it holds, e.g., if $$i=\dim M$$, $$i=1$$, and if g has ’normal decay of volume’. This does not work for cup products of 1-dimensional classes, however, leading to an open problem concerning intersection theory for foliations with non compact leaves.
Reviewer: H.Crauel

MSC:
 37A99 Ergodic theory 28D20 Entropy and other invariants 54H20 Topological dynamics (MSC2010) 54C70 Entropy in general topology
Keywords:
Shub’s entropy conjecture
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