On two occasions I allowed someone to get into my computer to resolve a
Problem. One person I know very well , no problem. The other was
Someone who owned a computer repair shop where I use to live.

When I sign on, the box for team viewer with the arrow going both ways
Has a black dot under it and it jumps up and down. Could this mean
Someone is getting into my computer.

The other black dot is my Safari... no other black dots.
I have a MAC air

No, teamviewer its very secure in both ends, when initialized you will notice by the change of resolution in your screen and chatbox will open. The program its compilated and works through the servers so it will be so difficult to get into your computer in this way. There are more efficient ways to get into your computer like trojans and so and you will never notice...

I understand your response and it makes sense. Still, do you know why the
Team viewer button jumps up and down and has a black dot under it?