Overview
This work investigates the dynamical behavior of surjective rational maps operating on smooth projective surfaces. The primary focus involves identifying and characterizing several dynamical properties pertinent to these maps. Additionally, the study introduces a numerical criterion intended to determine the regularity of such rational maps. Practical demonstrations include explicit constructions and calculations, specifically addressing aspects related to the topological entropy associated with these maps.
Research Context
The study centers on surjective rational maps, denoted as $f: X \dashrightarrow X$. The domain and codomain, $X$, are defined as smooth projective surfaces. The core mathematical constructs under examination are the dynamical degree of these maps and their regularity. The topological entropy of $f$ is also a component of the investigations, with explicit calculations provided within the context of the study.
Approach
The research approach involves establishing specific dynamical properties for the aforementioned class of maps. A key methodological element is the derivation of a numerical condition. This condition serves as a diagnostic tool for assessing the regularity of the rational map $f$. To support and illustrate these theoretical findings, the authors include explicit constructions and calculations. These computational examples are specifically linked to the topological entropy of the map $f$, providing concrete instances of the concepts discussed.
Findings
- Specific dynamical properties were established for surjective rational maps $f: X \dashrightarrow X$ where $X$ represents smooth projective surfaces.
- A numerical condition for the regularity of $f$ was provided.
- Explicit constructions and calculations were presented, which are related to the topological entropy of $f$.