On the dynamics of a semigroup and its relation with the Riemann Hypothesis
Carlos F. Álvarez, Juan Manzur
TL;DR
This work analyzes the semigroup of weighted composition operators $\mathcal{W}=(W_n)$ on the Hardy space $H^2$, focusing on the adjoints $W_n^*$ for $n\ge2$. It proves that each $W_n^*$ is mixing, Devaney chaotic, and frequently hypercyclic, using eigenvector density results and the Godefroy–Shapiro criterion, and it connects these dynamical properties to the Riemann Hypothesis (RH) and the invariant-subspace problem (ISP). A new RH-equivalence is established: RH holds iff there exists a hypercyclic vector for some $W_k^*$ whose orbit eventually lies in the closure of the subspace $\mathcal{N}=\operatorname{span}\{h_k\}$. The results illuminate how linear-dynamic properties of $W_n^*$ interact with deep number-theoretic questions, and they provide a new perspective on invariant subspaces in this setting.
Abstract
The semigroup of weighted composition operators $(W_n)_{n\in \mathbb{N}}$, defined by $$W_nf(z)=(1+z+\cdots +z^n)f(z^n),$$ acts on the classical Hardy-Hilbert space $H^{2}(\mathbb{D})$, and exhibits intriguing connections with both the Riemann Hypothesis (RH) and the Invariant Subspace Problem (ISP). In this paper, we prove that the adjoint operators $W^{\ast}_{n}$, for $n\geq 2$, are Devaney chaotic, frequently hypercyclic and mixing. In particular, these operators are hypercyclic and discuss connections with the RH and invariant subspaces.