Optimal growth of upper frequently hypercyclic functions for some weighted Taylor shifts

TL;DR

This work analyzes how rapidly hypercyclic and related vectors can grow (in $L^p$-averages) under weighted Taylor shifts $T_{\alpha}$ on the disk algebra $H(\mathbb{D})$. It introduces a continuous family of intermediate visit frequencies via $\mathcal{U}_{\beta^{\gamma}}$ densities that interpolate between hypercyclicity and $\mathcal{U}$-frequent hypercyclicity, and derives precise growth thresholds (critical exponents) depending on $p$ and $\gamma$. The authors provide both upper and lower bounds, including constructive proofs based on Rudin–Shapiro polynomials, showing optimality except at critical exponents where growth can be made arbitrarily slow. They furthermore treat the critical-exponent case, proving that growth can be slowed without bound for both $\mathcal{U}$- and $\mathcal{U}_{\beta^{\gamma}}$-frequent hypercyclicity, thereby linking the rate of growth to the chosen density. Overall, the paper unifies and extends the understanding of boundary behavior and growth rates for hypercyclic dynamics of weighted Taylor shifts.

Abstract

We are interested in the optimal growth in terms of $L^p$-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic function on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between the $\mathcal{U}$-frequent hypercyclicity and the hypercyclicity.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof