Entropy and twisted cohomology
type: Item
instance of: scholarly article
title: Entropy and twisted cohomology
zbMATH Open document ID: 0611.58036
DOI: 10.1016/0040-9383(86)90024-8
author: David Fried
published in: Topology
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54C70
zbMATH DE Number: 3988322
zbMATH Keywords: Shub's entropy conjecture
reviewed by: Hans Crauel
MaRDI profile type: Publication
full work available at URL: 0040-9383(86)90024-8
OpenAlex ID: W2029993683
description: scientific article; zbMATH DE number 3988322