This is a purity result for a polarized Kähler dynamical system. The precise statement is that if $(X,\omega)$ is compact Kähler and $\phi : X \to X$ an endomorphism having the class $[\omega] \in H^2(X,\mathbb{R})$ as eigenvector of eigenvalue $q > 1$ (a polarized Kähler dynamical system of positive entropy), then all eigenvalues for the action of $\phi^*$ on $H^i(X,\mathbb{R})$ have modulus $\sqrt{q}^i$.

A number of consequences of this result for algebraic and Kähler dynamics are derived by Shou-wu Zhang in [Distributions in algebraic dynamics, Surveys in Differential Geometry, vol. X, 2006], which also reproduces the proof of Serre's theorem, in mildly generalized form.

As for the zeta function, not explicitly considered in Serre's paper, it is a special case of a dynamical zeta, which is a vast subject.

Incidentally, there is something else going by the name of Kähler zeta function, discussed in Anton Deitmar's article A panorama of zeta functions from the volume [Erich Kähler. Mathematische Werke., De Gruyter, 2003], also available on ArXiv. Kähler's zeta function is actually arithmetical; it is a different idea of generalizing Riemann's zeta to finitely generated fields than the much better known (and better behaved) Hasse-Weil zeta function. Kähler's interest in finitely generated fields is well known (suffice it to recall the notion of the Kähler differential), but I was amused to learn from J.-B. Bost's article (A neglected aspect of Kähler's work in arithmetic geometry) in this volume that the modern subject of Arithmetic Geometry apparently takes its name from the title of a paper (Geometria Aritmetica) that Kähler wrote in Italian, and that Kähler's writings contain the germs of ideas and notions that later found their proper place in Arakelov theory, such as the Faltings height of an abelian variety.

This is a purity result for a polarized Kähler dynamical system. The precise statement is that if $(X,\omega)$ is compact Kähler and $\phi : X \to X$ an endomorphism having the class $[\omega] \in H^2(X,\mathbb{R})$ as eigenvector of eigenvalue $q > 1$ (a polarized Kähler dynamical system of positive entropy), then all eigenvalues for the action of $\phi^*$ on $H^i(X,\mathbb{R})$ have modulus $\sqrt{q}^i$.

A number of consequences of this result for algebraic and Kähler dynamics are derived by Shou-wu Zhang in [Distributions in algebraic dynamics, Surveys in Differential Geometry, vol. X, 2006], which also reproduces the proof of Serre's theorem in mildly generalized form.

As for the zeta function, not explicitly considered in Serre's paper, it is a special case of a dynamical zeta, which is a vast subject.

Incidentally, there is something else going by the name of Kähler zeta function, discussed in Anton Deitmar's article A panorama of zeta functions from the volume [Erich Kähler. Mathematische Werke., De Gruyter, 2003], also available on ArXiv. Kähler's zeta function is actually arithmetical; it is a different idea of generalizing Riemann's zeta to finitely generated fields than the much better known (and better behaved) Hasse-Weil zeta function. Kähler's interest in finitely generated fields is well known (suffice it to recall the notion of the Kähler differential), but I was amused to learn from J.-B. Bost's article in that volume (A neglected aspect of Kähler's work in arithmetic geometry) that the modern subject of Arithmetic Geometry apparently takes its name from the title of a paper (Geometria Aritmetica) that Kähler wrote in Italian, and that Kähler's writings contain the germs of ideas and notions that later found their proper place in Arakelov theory, such as the Faltings height of an abelian variety.

This is a purity result for a polarized Kähler dynamical system. The precise statement is that if $(X,\omega)$ is compact Kähler and $\phi : X \to X$ an endomorphism having the class $[\omega] \in H^2(X,\mathbb{R})$ as eigenvector of eigenvalue $q > 1$ (a polarized Kähler dynamical system of positive entropy), then all eigenalues for the action of $\phi^*$ on $H^i(X,\mathbb{R})$ have eigenvalue $\sqrt{q}^i$.

A number of consequences of this result for algebraic and Kähler dynamics are derived by Shou-wu Zhang in [Distributions in algebraic dynamics, Surveys in Differential Geometry, vol. X, 2006], which also contains the proof of a slightly generalized form of Serre's theorem.

As for the zeta function, not explicitly considered in Serre's paper, it is a special case of a dynamical zeta, which is a vast subject.

Incidentally, there is something else going by the name of Kähler zeta function, discussed in Anton Deitmar's article A panorama of zeta functions from the volume [Erich Kähler. Mathematische Weke., De Gruyter, 2003], also available on ArXiv. Kähler's zeta function is actually arithmetical; it is a different idea of generalizing Riemann's zeta to finitely generated fields than the much better known (and better behaved) Hasse-Weil zeta function. Kähler's interest in finitely generated fields is well known (suffice it to recall the notion of the Kähler differential), but I was amused to learn from J.-B. Bost's article (A neglected aspect of Kähler's work in arithmetic geometry) in this volume that the name of the modern subject of Arithmetic Geometry derives from a paper (Geometria Aritmetica) that Kähler wrote in Italian, and that Kähler's writings contain the germs of notions that later found their proper place in Arakelov theory, such as the Faltings height of an abelian variety.

This is a purity result for a polarized Kähler dynamical system. The precise statement is that if $(X,\omega)$ is compact Kähler and $\phi : X \to X$ an endomorphism having the class $[\omega] \in H^2(X,\mathbb{R})$ as eigenvector of eigenvalue $q > 1$ (a polarized Kähler dynamical system of positive entropy), then all eigenvalues for the action of $\phi^*$ on $H^i(X,\mathbb{R})$ have modulus $\sqrt{q}^i$.

A number of consequences of this result for algebraic and Kähler dynamics are derived by Shou-wu Zhang in [Distributions in algebraic dynamics, Surveys in Differential Geometry, vol. X, 2006], which also reproduces the proof of Serre's theorem, in mildly generalized form.

As for the zeta function, not explicitly considered in Serre's paper, it is a special case of a dynamical zeta, which is a vast subject.

Incidentally, there is something else going by the name of Kähler zeta function, discussed in Anton Deitmar's article A panorama of zeta functions from the volume [Erich Kähler. Mathematische Werke., De Gruyter, 2003], also available on ArXiv. Kähler's zeta function is actually arithmetical; it is a different idea of generalizing Riemann's zeta to finitely generated fields than the much better known (and better behaved) Hasse-Weil zeta function. Kähler's interest in finitely generated fields is well known (suffice it to recall the notion of the Kähler differential), but I was amused to learn from J.-B. Bost's article (A neglected aspect of Kähler's work in arithmetic geometry) in this volume that the modern subject of Arithmetic Geometry apparently takes its name from the title of a paper (Geometria Aritmetica) that Kähler wrote in Italian, and that Kähler's writings contain the germs of ideas and notions that later found their proper place in Arakelov theory, such as the Faltings height of an abelian variety.