Frequently hypercyclic and Devaney chaotic $ C_{0} $-semigroups indexed by complex sectors

Shengnan He; Zongbin Yin; Shengnan He; Zongbin Yin

doi:10.3934/math.2025915

AIMS Mathematics

2025, Volume 10, Issue 9: 20505-20530. doi: 10.3934/math.2025915

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Research article

Frequently hypercyclic and Devaney chaotic $ C_{0} $-semigroups indexed by complex sectors

Shengnan He ¹,
Zongbin Yin ^{2
,
,}

1.
School of Humanities and Fundamental Sciences, Shenzhen University of Information Technology, Shenzhen, 518172, China
2.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou, 510665, China

Received: 03 July 2025 Revised: 27 August 2025 Accepted: 29 August 2025 Published: 05 September 2025
MSC : 37B05, 37B20, 47A16, 47D03

In this paper, the definitions of frequent hypercyclicity and Devaney chaos for a $ C_0 $-semigroup $\left\{T_{t}\right\}_{t\in\Delta}$ indexed by a complex sector $ \Delta $ are revisited. A general sufficient criterion and necessary conditions for a $ C_0 $-semigroup $\left\{T_{t}\right\}_{t\in\Delta}$ to be frequently hypercyclic are established. By adapting the concept of periodic point with a syndetic set of periods, we present clean and effective characterizations of the Devaney chaotic translation semigroup $\left\{T_{t}\right\}_{t\in\Delta}$ on $ L^{p}_{\rho}(\Delta, \mathbb{K}), 1\leq p < \infty $, which generalize those of the classical translation semigroup $\left\{T_{t}\right\}_{t \geq 0}$. Moreover, it is shown that Devaney chaos implies frequent hypercyclicity, topological mixing, and distributional chaos for the translation semigroups $\left\{T_{t}\right\}_{t\in\Delta}$ under our revised definitions, and that topological mixing or distributional chaos does not imply Devaney chaos.
- frequent hypercyclicity,
- Devaney chaos,
- complex sector,
- $ C_0 $-semigroups,
- translation semigroup,
- periodic point
Citation: Shengnan He, Zongbin Yin. Frequently hypercyclic and Devaney chaotic $ C_{0} $-semigroups indexed by complex sectors[J]. AIMS Mathematics, 2025, 10(9): 20505-20530. doi: 10.3934/math.2025915

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Abstract

In this paper, the definitions of frequent hypercyclicity and Devaney chaos for a $ C_0 $-semigroup $\left\{T_{t}\right\}_{t\in\Delta}$ indexed by a complex sector $ \Delta $ are revisited. A general sufficient criterion and necessary conditions for a $ C_0 $-semigroup $\left\{T_{t}\right\}_{t\in\Delta}$ to be frequently hypercyclic are established. By adapting the concept of periodic point with a syndetic set of periods, we present clean and effective characterizations of the Devaney chaotic translation semigroup $\left\{T_{t}\right\}_{t\in\Delta}$ on $ L^{p}_{\rho}(\Delta, \mathbb{K}), 1\leq p < \infty $, which generalize those of the classical translation semigroup $\left\{T_{t}\right\}_{t \geq 0}$. Moreover, it is shown that Devaney chaos implies frequent hypercyclicity, topological mixing, and distributional chaos for the translation semigroups $\left\{T_{t}\right\}_{t\in\Delta}$ under our revised definitions, and that topological mixing or distributional chaos does not imply Devaney chaos.

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