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Christopher Deninger

Publications and source records attributed to Christopher Deninger.

33 records · Page 2Linked to original sources
arXiv2011

Invariant measures on the circle and functional equations

It is well known that real measures on the circle are characterized by their Herglotz transform, an analytic function in the unit disc. Invariance of the measure under N-multiplication translates into a functional equation for the Herglotz transform. Using elements from the theory of Hardy spaces one gets a somewhat surprising condition for a sequence of complex numbers to be the Fourier coefficients of an N-invariant measure. Next, starting from any atomless measure on the circle we construct atomless premesures of bounded kappa-variation in the sense of Korenblum which are invariant under s given pairwise prime integers. The relevant function kappa is a generalized entropy function depending on s. The proof uses Korenblum's generalized Nevanlinna theory. Passing to "kappa-singular measures" and extending these to elements in a Grothendieck group of possibly unbounded measures on the circle, one obtains generalized invariant measures which are carried by "kappa-Carleson" sets. The range of this construction depends on interesting questions about cyclicty in growth algebras of analytic functions on the unit disc. We also describe some very formal relations with Witt vectors. For example the Artin-Hasse p-exponential "is" a p-invariant premeasure of bounded kappa_1 variation.

math.DS↗
arXiv2011

Regulators, entropy and infinite determinants

In this note we describe instances where values of the $K$-theoretical regulator map evaluated on topological cycles equal entropies of topological actions by a group $Γ$. These entropies can also be described by determinants on the von Neumann algebra of $Γ$. The relations were first observed for real regulators. The latter have $p$-adic analogues and both $p$-adic entropy and $p$-adic determinants were then defined so that similar relations hold as in the real case. We describe this $p$-adic theory in the second part of the paper. This note is almost entirely a survey of known results with the exception of some results in section 3.1. However the different aspects of the theory have not been discussed together before. Along the way we point out several open questions and possible directions for further research.

math.AG↗
arXiv2009

Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents

For an ergodic measure preserving action on a probability space, consider the corresponding crossed product von Neumann algebra. We calculate the Fuglede-Kadison determinant for a class of operators in this von Neumann algebra in terms of the Ljapunov exponents of an associated measurable cocycle. The proof is based on recent work of Dykema and Schultz. As an application one obtains formulas for the Fuglede-Kadison determinant of noncommutative polynomials in the von Neumann algebra of the discrete Heisenberg group. These had been previously obtained by Lind and Schmidt via entropy considerations.

math.OA↗
arXiv2007

A dynamical systems analogue of Lichtenbaum's conjectures on special values of Hasse-Weil zeta functions

In recent years Lichtenbaum has conjectured a description for the special values of Hasse--Weil zeta functions in terms of ``Weil-étale cohomology''. In earlier papers we studied a class of foliated dynamical systems which had some similarities with arithmetic schemes. Assuming that certain zeta regularized determinants exist we now prove an analogue of Lichtenbaum's conjectures for a particularly simple class of such dynamical systems: Using results of Álvarez López and Kordyukov we express the leading coefficient $ζ^*_R (0)$ of the Ruelle zeta function $ζ_R (s)$ at $s = 0$ in terms of analytic torsion. We then apply the Cheeger--Müller theorem to replace the analytic torsion by the Reidemeister torsion with respect to harmonic bases. For our dynamical systems the latter can be expressed by the same recipe as the one in Lichtenbaum's conjectures.

math.NT↗
arXiv2006

Expansive algebraic actions of discrete residually finite amenable groups and their entropy

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $Γ$ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $Γ$-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $Γ_n$-fixed points, where $(Γ_n, n\ge1)$ is a decreasing sequence of finite index normal subgroups of $Γ$ with trivial intersection.

math.DS↗
arXiv2005

Vector bundles on p-adic curves and parallel transport

We define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.

math.AG↗
arXiv2005

Fuglede-Kadison determinants and entropy for actions of discrete amenable groups

In 1990, Lind, Schmidt and Ward gave a formula for the entropy of certain $\mathbb{Z}^n$-dynamical systems attached to Laurent polynomials $P$, in terms of the (logarithmic) Mahler measure of $P$. We extend the expansive case of their result to the noncommutative setting where $\mathbb{Z}^n$ gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede-Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.

math.DS↗
arXiv2004

Line bundles and p-adic characters

For a certain class of vector bundles E on abelian varieties A over local fields containing all line bundles algebraically equivalent to zero we define a canonical representation of the Tate module of A on the fibre of E in the zero section. This extends an old construction of Tate for line bundles to vector bundles of higher rank. We also compare this construction to the theory of parallel transport for vector bundles on p-adic curves developed in mathAG/0403516. Relations with the Hodge-Tate decomposition are also explained.

math.AG↗
arXiv2002

Two-variable zeta functions and regularized products

In this paper we prove a regularized product expansion for the two-variable zeta functions of number fields introduced by van der Geer and Schoof. The proof is based on a general criterion for zeta-regularizability due to Illies. For number fields of non-zero unit rank our method involves a result of independent interest about the asymptotic behaviour of certain oscillatory integrals in the geometry of numbers. We also explain the cohomological motivation for the paper.

math.NT↗
arXiv2002

Exchanging the places p and infinity in the Leopoldt conjecture

The Leopoldt conjecture is concerned with the image of the global units in the local units at the primes dividing p. In the definition of the global units the infinite place is distinguished. Exchanging p and infinity in the formulation one gets a new conjecture. It predicts that certain vectors should be linearly independent over the reals whose components are arguments of conjugates of Weil numbers. Using Baker's result on linear forms in logarithms we prove part of this new conjecture in certain abelian situations.

math.NT↗
arXiv2002

A dynamical Lefschetz trace formula for algebraic Anosov diffeomorphisms

For algebraic Anosov diffeomorphisms we first express the reduced leafwise cohomology with respect to the unstable foliation in terms of finite dimensional Lie algebra cohomology. We then prove a dynamical Lefschetz trace formula for the induced action on this cohomology using a result of Nomizu. We also consider generalized Anosov maps for which the cohomology in question is no longer finite dimensional in general. Using the representation theory of nilpotent Lie groups we still arrive at a meaningful definition of traces on cohomology. Again we establish a dynamical Lefschetz trace formula.

math.DS↗
arXiv2002

A note on arithmetic topology and dynamical systems

We discuss analogies between the etale site of arithmetic schemes and the algebraic topology of dynamical systems. The emphasis is on Lefschetz numbers. We also discuss similarities between infinite primes in arithmetic and fixed points of dynamical systems.

math.NT↗
arXiv2002

Number theory and dynamical systems on foliated spaces

We discuss analogies between number theory and the theory of dynamical systems on spaces with a one-codimensional foliation. The emphasis is on comparing the "explicit formulas" of analytic number theory with certain dynamical Lefschetz trace formulas. We also point out a possible relation between an Arakelov-Euler characteristic and an Euler characteristic in the sense of Connes. Finally the role of generalized solenoids as phase spaces in our picture is explained.

math.NT↗
arXiv2002

Real polarizable Hodge structures arising from foliations

We construct real polarizable Hodge structures on the reduced leafwise cohomology of Kähler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's Kählerian analogue of the Weil conjectures carries over as well. Generalizing a construction of Looijenga and Lunts one obtains possibly infinite dimensional Lie algebras attached to Kähler-Riemann foliations. Finally using $(\mathfrak{g},K)$-cohomology we discuss a class of examples obtained by dividing a product of symmetric spaces by a cocompact lattice and considering the foliations coming from the factors.

math.DG↗