# A snapshot of holomorphic dynamics in a neighborhood of a fixed point

## Document Type

Presentation Abstract

## Presentation Date

2-18-2016

## Abstract

Holomorphic dynamics is a fascinating area of mathematics that lies at the intersection of complex analysis and dynamical systems. In pop-culture, holomorphic dynamics is known for its beautiful fractals, like the Mandelbrot set and Julia sets. This talk will be a gentle introduction to holomorphic dynamics and, along the way, we will see and explore many beautiful fractals.

More specifically, we will focus on holomorphic self-maps that fix a point and explore the dynamics near that fixed point. Are nearby points attracted to (or repelled from) that fixed point when the map is iterated? If so, how? These questions are of great interest in holomorphic dynamics in one (and several) complex variables. We will begin our discussion in dimension one and reveal the local dynamics near a fixed point of a general holomorphic map. We will see how the coefficient of the linear term in the power series expansion of the map near the fixed point can determine how points near that fixed point behave under iteration. We will then focus on a particularly interesting class of maps that are called tangent to the identity. In dimension one, the Leau-Fatou Flower Theorem provides a beautiful description of the behavior of points in a full neighborhood of a fixed point for this class of maps. This theorem from the early 1900s serves as inspiration for the study of maps tangent to the identity in higher dimensions. In higher dimensions, our picture of the dynamics near a fixed point is still being formed. We will briefly introduce the study of maps tangent to the identity in several complex variable and we will establish the foundations for our discussion in the subsequent analysis seminar.

## Recommended Citation

Lapan, Sara, "A snapshot of holomorphic dynamics in a neighborhood of a fixed point" (2016). *Colloquia of the Department of Mathematical Sciences*. 503.

https://scholarworks.umt.edu/mathcolloquia/503

## Additional Details

Wednesday, February 17, 2016 at 2:10 p.m. in Math 103

Refreshments at 3:00 p.m. in Math Lounge 109