On the dynamics of a semigroup and its relation with the Riemann Hypothesis

TL;DR

This paper explores the properties of a semigroup of weighted composition operators on Hardy space, revealing their chaotic behavior and potential links to the Riemann Hypothesis and invariant subspace problem.

Contribution

It demonstrates that the adjoint operators are chaotic, hypercyclic, and mixing, establishing new connections between operator dynamics and fundamental mathematical conjectures.

Findings

Operators are Devaney chaotic for n ≥ 2

Operators are frequently hypercyclic and mixing

Links between operator dynamics, RH, and ISP

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.