Semi-horseshoe and Expanding Bundles

Key Points

• This work aims to explore the semi-horseshoe structure and its relationship with entropy within dynamical systems.
• Review of semi-horseshoe existence results on compact smooth Riemann manifolds with nontrivial expanding bundles.
• Analysis of the quantitative relationship between semi-horseshoes, expansion sub-bundles, and their logarithmic growth rates.
• Discussion of the connection between main semi-horseshoes and intermediate entropy properties in topological dynamical systems.
• Establishment of existence results for semi-horseshoes in $C^1$ diffeomorphisms on compact smooth Riemann manifolds.
• Quantitative relation proves the entropy approximations from the contributions of multiple expansion directions.
• Findings indicate that if a dynamical system has a series of main semi-horseshoes, its entropy can take any value in the interval $[0,h_{top}(f))$.