Probably one of the variegated forms of Euonymus fortunei. Planted under a birch or fig any issue with water sharing that might arise would be due to the trees taking all the water away from the ground cover planting, and not the other way around.

In my area this plant is prone to weevils and would be likely to be damaged noticeably beneath a birch.

invasive in Virginia

you seem to worry the tree will be affected ... but its mostly the other way around ...

the trees will be super competitors ... and could possible dominate the available water ...over time ...

is anything else growing under your trees ??? ... there may be a reason why the area is empty ....

i dont hit embedded links.. you can put the pix in posts.. by using the camera/photo icon below the box you type in ...

also.. add your location to all posts ... it really matters ... as noted from the replies above ....

ken

And with variegated vinca, variegated pachysandra and if you are really brave and with confined area like the photo, variegated goutweed.

If you want possible alternatives, let us know generally where you are for the best suggestions.

Thanks so much guys! We are in zone 7. I would like something that puts some ground cover, evergreen would be great, and something that can't attach itself to brick. Something that self attached we don't want.

Thanks so much!

We need a state too. Zone 7 can be in New Mexico or New Jersey. Big difference.

Sun or shade? Direction it faces?

There are a couple of photos here, though hopefully midorit will add some clearer photos.

The Euonymus and ivy/Hedera will attach itself to brick.

I grow cranberry as a groundcover, but I don't know how it would do in zone 7, if I recall from your other posts, the mid-Atlantic or DC area. It doesn't need a bog, just ordinary garden soil.

Other groundcovers to check out, along with those that GG48 suggested:

Lamium/dead nettle

Galium/bedstraw

European Ginger

Epimedium

Dwarf Solomon's seal

Veronica 'Georgia Blue'

Cranberry and the last 4 are generally relatively sedate spreaders, while Lamium, Galium, and GG48's are quite vigorous.