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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
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