[Paper Review] Invariant measures on the circle and functional equations

Christopher Deninger|arXiv (Cornell University)|Nov 28, 2011

Mathematical Dynamics and Fractals10 references3 citations

TL;DR

This paper studies $φ_N$-invariant probability measures on the unit circle $×$ via functional equations in holomorphic functions, particularly in the Nevanlinna class and generalized Hardy spaces. It establishes that non-constant solutions to the functional equation $f(z^N)^N = \prod_{\zeta^N=1} f(\zeta z)$ are quotients of singular inner functions, and characterizes simultaneous invariance under multiple endomorphisms $\varphi_N$ using a novel isomorphism between solution spaces and premeasures of bounded $\kappa_s$-variation.

ABSTRACT

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.

Motivation & Objective

• To characterize $\varphi_N$-invariant probability measures on the unit circle $\mathbb{T}$ using functional equations in holomorphic functions.
• To understand the structure of solutions to the functional equation $f(z^N)^N = \prod_{\zeta^N=1} f(\zeta z)$ in the Nevanlinna class $\mathcal{N}$.
• To extend the analysis to simultaneous invariance under multiple endomorphisms $\varphi_N$ and $\varphi_M$ with $N$ coprime to $M$, using the monoid generated by such $N$.
• To relate the solutions to generalized premeasure spaces of bounded $\kappa_s$-variation, particularly for $s=1$, and to characterize their analytic counterparts via the $\Psi_{{\mathcal{S}}}$ and $\Phi_{{\mathcal{S}}}$ maps.
• To show that certain entire functions like $E_N = \exp(\sum_{\nu=0}^\infty z^{N^\nu}/N^\nu)$ arise as Weierstrass transforms of non-measure premeasures, and to prove their $p$-integrality and idempotency.

Proposed method

• The paper uses the Herglotz transform $h_\mu(z) = \int_{\mathbb{T}} \frac{\zeta + z}{\zeta - z} d\mu(\zeta)$ to associate a holomorphic function $f_\mu = \exp(-h_\mu)$ to each measure $\mu$ on $\mathbb{T}$, linking measure invariance to functional equations.
• It establishes that $\varphi_N$-invariance of $\mu$ is equivalent to the functional equation $f(z^N)^N = \prod_{\zeta^N=1} f(\zeta z)$ for $f = f_\mu$ in the Nevanlinna class $\mathcal{N}$.
• The solution space $H^0(\mathcal{S}, \mathcal{O}^1)$ of functions satisfying the functional equation for all $N$ in a multiplicative semigroup $\mathcal{S}$ of pairwise coprime integers is studied via the mutually inverse isomorphisms $\Psi_{\mathcal{S}}$ and $\Phi_{\mathcal{S}}$, defined by $\Phi_{\mathcal{S}}(f)(z) = f(z)/f(z^N)$ and $\Psi_{\mathcal{S}}(\alpha)(z) = \prod_{\nu=0}^\infty \alpha(z^{N^\nu})$.
• The paper introduces generalized growth classes $\mathcal{N}_s$ defined by growth estimates $|g(z)| \leq a_g \exp(r_g \log^s(1 - |z|)^{-1})$, and shows that $\Psi_{\mathcal{S}}(Z(\mathcal{S}, \mathcal{N}^1)) \subset H^0(\mathcal{S}, \mathcal{N}^1_s)$, linking analytic growth to measure singularities.
• It applies Korenblum's theory to identify elements of $\mathcal{N}^1_s$ with real premeasures of bounded $\kappa_s$-variation, where $\kappa_s(x) = x \sum_{\nu=0}^s \frac{1}{\nu!} |\log x|^\nu$, and uses this to characterize $\varphi_N$-invariant premeasures.
• The construction of the function $E_N = \exp(\sum_{\nu=0}^\infty z^{N^\nu}/N^\nu)$ as a Weierstrass transform of a non-measure premeasure $\mu$ is central, with $w(\mu) = E_N$, and the proof that $\mu$ is atomless, $N$-invariant, idempotent, and not a signed measure.

Experimental results

Research questions

• RQ1 Which holomorphic functions in the Nevanlinna class $\mathcal{N}$ satisfy the functional equation $f(z^N)^N = \prod_{\zeta^N=1} f(\zeta z)$ for a given $N \geq 2$, and what is their structure?
• RQ2 How can the space of solutions to the functional equation be characterized for a multiplicative semigroup $\mathcal{S}$ generated by pairwise coprime integers $N_1, \dots, N_s$, and what is the role of the $\Psi_{\mathcal{S}}$ and $\Phi_{\mathcal{S}}$ isomorphisms?
• RQ3 What is the relationship between the growth of holomorphic functions in $\mathcal{N}^1_s$ and the $\kappa_s$-variation of associated premeasures on the circle?
• RQ4 Can the function $E_N = \exp(\sum_{\nu=0}^\infty z^{N^\nu}/N^\nu)$ be realized as the Weierstrass transform of a premeasure, and what are the properties of this premeasure?
• RQ5 Is there a non-measure premeasure $\mu$ on $\mathbb{T}$ such that $\mu$ is $\varphi_N$-invariant, $\mu * \mu = \mu$, and $w(\mu) = E_N$?

Key findings

• Any non-zero function $f \in \mathcal{N}$ satisfying the functional equation $f(z^N)^N = \prod_{\zeta^N=1} f(\zeta z)$ is a quotient of singular inner functions; Blaschke products and outer functions in $\mathcal{N}$ cannot satisfy the equation unless constant.
• The map $\Psi_{\mathcal{S}}: Z(\mathcal{S}, \mathcal{N}^1) \to H^0(\mathcal{S}, \mathcal{N}^1_s)$ is well-defined and satisfies $\Psi_{\mathcal{S}}(Z(\mathcal{S}, \mathcal{N}^1)) \subset H^0(\mathcal{S}, \mathcal{N}^1_s)$, with $\mathcal{N}^1_s$ defined by growth estimates $|g(z)| \leq a_g \exp(r_g \log^s(1 - |z|)^{-1})$.
• For $s=1$, the class $\mathcal{N}^1_1$ corresponds to functions of exponential type $O((1 - |z|)^{-r})$, and is identified with premeasures of bounded $\kappa_1$-variation, where $\kappa_1(x) = x(1 + |\log x|)$.
• The function $E_N = \exp(\sum_{\nu=0}^\infty z^{N^\nu}/N^\nu)$ is $p$-integral and idempotent under the circle product $\odot$, and arises as $w(\mu)$ for a unique premeasure $\mu$ on $\mathbb{T}$ with $\mu \in P^+_{\kappa_1}(\mathbb{T}) \cap P^-_{\kappa_1}(\mathbb{T})$, $N_*\mu = \mu$, and $\mu * \mu = \mu$.
• The premeasure $\mu$ is atomless, $\varphi_N$-invariant, and not a signed measure; its Herglotz transform is $h_\mu(z) = -2 \sum_{\nu=0}^\infty z^{N^\nu}$, and $f_\mu = \exp(-h_\mu) = \prod_{\nu=0}^\infty \exp(2z^{N^\nu})$ is a unit in $\mathcal{A}_1$ but not in $\mathcal{N}$.
• The $\kappa_1$-singular measure $\mu_s = \sigma_{f_\mu}$ vanishes, and the measure $\mu$ cannot be a signed measure because its associated function $\sigma = \Phi_{\mathcal{S}}(\mu)$ is absolutely continuous, contradicting the singularity required for a measure.