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Dynamics of the functions [Formula: see text] with the real parameter
Xiaocheng Deng1, Fanning Meng1, Jianming Lin2
1School of Mathematics and Information Science, Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, 510006 People's Republic of China.
Springerplus
|July 8, 2016
Summary
This study explores the dynamics of functions with real parameters, proving the Fatou set is an attracting basin. It also details properties of the set for specific parameters, including its non-emptiness and component structure.
Area of Science:
- Dynamical systems
- Complex analysis
- Real parameter dynamics
Background:
- The study investigates the behavior of functions under real parameter variations.
- Understanding the Fatou set's properties is crucial for analyzing dynamical systems.
Purpose of the Study:
- To analyze the dynamics of functions with a real parameter.
- To characterize the Fatou set and specific parameter sets within these dynamics.
Main Methods:
- Analysis of function dynamics under real parameter influence.
- Proof of the Fatou set's properties as a completely invariant attracting basin.
- Investigation of the structure and components of specific parameter sets.
Main Results:
- The Fatou set is proven to be a completely invariant attracting basin for all parameters.
- Existence of a parameter for which a specific condition holds.
- Demonstration that for any positive integer n, a related set is non-empty.
- Proof that for any prime number p, a related set contains at least two components.
Conclusions:
- The Fatou set exhibits robust attracting basin properties across all parameters.
- Specific parameter sets possess non-trivial structures, including multiple components for prime-related sets.
- These findings contribute to a deeper understanding of real parameter dynamics in complex functions.
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