We investigate contrasting behaviours emerging when studying foliations on non-metrisable manifolds. It is shown that Kneser's pathology of a manifold foliated by a single leaf cannot occur with foliations of dimension-one. On the other hand, there are open surfaces admitting no foliations. This is derived from a qualitative study of foliations defined on the long tube S 1 × L + (product of the circle with the long ray), which is reminiscent of a 'black hole', in as much as the leaves of such a foliation are strongly inclined to fall into the hole in a purely vertical way. More generally the same qualitative behaviour occurs for dimension-one foliations on M × L + , provided that the manifold M is "sufficiently small", a technical condition satisfied by all metrisable manifolds. We also analyse the structure of foliations on the other of the two simplest long pipes of Nyikos, the punctured long plane. We are able to conclude that the long plane L 2 has only two foliations up to homeomorphism and six up to isotopy.

We generalize to tree maps the theorems of Parry and Milnor–Thurston about the semi-conjugacy of a continuous piecewise monotone map f to a continuous piecewise linear map with constant slope, equal to the exponential of the entropy of f.

We analyze the relationship between Hausdorffness and homogeneity in the frame of manifolds not confined to be Hausdorff. We exhibit examples of homogeneous non-Hausdorff manifolds and prove that a Lindelöf homogeneous manifold is Hausdorff.

Link to this article: http://journals.cambridge.org/abstract_S014338570500012XHow to cite this article: MATHIEU BAILLIF and VIVIANE BALADI (2005). Kneading determinants and spectra of transfer operators in higher dimensions: the isotropic case.Abstract. Transfer operators M k are associated to C r transversal local diffeomorphisms ψ ω of R n , and C r compactly supported functions g ω . A formal trace tr # M, yields a formal Ruelle-Lefschetz determinant Det # (Id −zM). We use the Milnor-Thurston-Kitaev equality recently proved by Baillif to relate zeros and poles of Det # (Id −zM) with spectra of the transfer operators M k , under additional assumptions. As an application, we obtain a new proof of a result of Ruelle on the spectral interpretation of zeros and poles of the dynamical zeta function exp m≥1 (z m /m) f m (x)=x |det Df (x)| −1 for smooth expanding endomorphisms f .

We study a transfer operator M (k) associated to a family f !g of C 3 transversal local di eomorphisms of R n , with C 3 compactly supported weights g!, and let it act on k-forms in R n. Using the de nitions of sharp trace Tr # and at trace Tr , the following formula holds between formal power series: Det # (1?zM) = n k=0 Det (1?zM (k)) (?1) k. Following ideas of Kitaev, we de ne the kneading operators D k (z), which are kernel operators. Our main result is the equality in odd dimension Det # (1 ? zM) = n?1 k=0 Det (1 + D k (z)) (?1) k+1 as formal power series (the determinant Det is de ned by the trace Tr which is the usual trace of kernel operators). As a consequence, we obtain that the weighted Lefschetz zeta-function # (z) = 1=Det # (1 ? zM) has a positive radius of convergence. This (partially)generalisesresults obtainedby Baladi, Kitaev, Ruelle and Semmes in dimension 1, complex and real.