Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions

Abstract

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.

Dynamical determinants and spectrum for hyperbolic diffeomorphisms

“…Then the flat trace of (L m ) b is not zero, but it decays exponentially, arbitrarily fast [4]. 6 The operator D(z) = zMc(Id − zM b ) −1 can be viewed as a kneading operator, [6], [3]. parallel with the definition of Q p,q (T, g) in Section 1, replacing χ µ (g/ det(DT |E u )) by χ µ (T/ det(DT |E u )).…”

Kneading determinants and spectra of transfer operators in higher dimensions: the isotropic case

“…( 2.13) We shall not need the following result, although we shall exploit and revisit key ideas from its proof (see [Bai,Lemma 6.2]) in Lemma 6 below.…”

Anosov Flows and Dynamical Zeta Functions

“…In section 3 we construct a family of Banach spaces B p,q,ℓ , p ∈ N, q ∈ R + , as the closure of Ω ℓ 0,v (M) with respect to a suitable anisotropic norm so that the spaces B p,q,ℓ are an extension of the spaces in [30]. 6 Such spaces are canonically embedded in the space of currents (see Lemma 3.10).…”

“…operators as well as connected dynamical determinants to appropriate "flat traces" 3 of transfer operators (see [6,7,10]). On the other hand, [43,16] illustrated how this approach can be applied also to flows by showing that the resolvent of the generator of the flow is quasi compact on such spaces.…”