EXTENSION OF PLURISUBHARMONIC FUNCTIONS AND DYNAMICS OF POLYNOMIAL MAPPINGS

<p>Let $X$ be an algebraic subvariety of \mathbb Cⁿ and $\overline X$ be its closure in \mathbb Pⁿ. Coman-Guedj-Zeriahi proved that any plurisubharmonic function with logarithmic growth on $X$ extends to a plurisubharmonic function with logarithmic growth on \mathbb Cⁿ when the germs $(\overline X,a)$ in \mathbb Pⁿ are irreducible for all $a\in \overline X\setminus X.$ In this dissertation, we consider $X$ for which the germ $(\overline X,a)$ is reducible for some $a\in \overline X\setminus X$ and give a necessary and sufficient condition for $X$ so that any plurisubharmonic function with logarithmic growth on $X$ extends to a plurisubharmonic function with logarithmic growth on \mathbb Cⁿ.</p> <p> We also study a problem in complex dynamics. Quadratic automorphisms of \mathbb C³ are classified up to affine conjugacy into seven classes by Forn\ae ss and Wu. Five of these classes contain maps with interesting dynamics. For these maps, Coman and Forn\ae ss estimated the rates of escape of orbits to infinity and described the subsets of \mathbb C³ where they occur. Using these estimates, they constructed invariant measures for the maps in three of these classes. By the work of Coman on the fourth class later, the dynamics of the maps from the first four classes is completely understood. This dissertation focuses on the dynamics of maps from the fifth class: $H(x,y,z)= (xy+az, x²+by, x), a\neq0\neq b.$ </p> <p>We investigate the behaviors of $H$ at infinity and construct a dynamically interesting closed positive current of bidimension $(1,1)$.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2015

Author dc:creator

• Yazici, Ozcan

Contributors dc:contributor

• Dan Coman

• Dynamics of polynomial automorphisms
• Invariant measures
• Lelong class
• Pluripotential theory
• Physical Sciences and Mathematics

Repository record dc:identifier

https://surface.syr.edu/etd/259

OAI identifier oai:identifier

oai:surface.syr.edu:etd-1259