Horseshoes for a Class of Nonuniformly Expanding Random Circle Maps

Area of Science:

• Dynamical Systems
• Chaos Theory
• Stochastic Processes

Background:

• Non-uniform expansion in dynamical systems presents unique challenges.
• Understanding random dynamical systems is crucial for modeling complex phenomena.
• Horseshoe dynamics are fundamental to chaotic behavior.

Purpose of the Study:

• To introduce and define the concept of a random horseshoe.
• To establish sufficient conditions for the abundant existence of random horseshoes.
• To analyze these conditions in the context of specific random dynamical systems.

Main Methods:

• Development of a theoretical framework for random horseshoes.
• Analysis of one-dimensional random dynamical systems with non-uniform expansion.
• Application of methods to random circle endomorphisms with additive noise.

Main Results:

• A novel notion of random horseshoe is proposed.
• Sufficient conditions guaranteeing the abundant existence of random horseshoes are provided.
• The existence is confirmed for non-uniformly expanding random circle endomorphisms with bounded additive noise.

Conclusions:

• The proposed random horseshoe concept offers new insights into chaotic dynamics.
• The established conditions provide a pathway for identifying such structures.
• This work contributes to the understanding of random dynamical systems and their properties.