Mumu, the pigeon that opened a landscape in my heart with a twig

The story started two years ago with a blackbird who was walking each morning by my window. We used to put some treats on his way: a cherry or a few nuts...
Other birds were attracted: other blackbirds, a jay, a starling, tits... We put a greaseball for the tits when it was winter. And they were there one day, shivering with cold: a couple of shaggy pigeons!

I gave them some food, and of course they came back very constantly! At that time, I was not interested in pigeons, these "common and gluttonous birds". So I was disappointed as the other birds couldn't approach the window anymore as it was occupied by these two...

But the male one, Mumu, had such a funny and irresistible way of stucking his orange eye to the window to watch inside the room! A real little person!
The female one, LittleWhite, was very shy, but touching in the way she tried to overcome her fear.

We gradually became friends. I started to read and learn lots of things about pigeons in order to understand Mumu's behavior. I discovered that they are not really wild birds and how much they suffer in our hostile cities, and how interesting, beautiful, funny, mysterious and friendly pigeons are... Here in Paris, France, the policy is to forbid to feed them, so they all struggle for food as there is no natural food available for them.

Mumu has bad legs because of threads, it was already too late to remove them. But he lives fine with his stumps now, he fights with other pigeons and walks almost normaly. He just cannot perch on trees...
LittleWhite, later on, had also strings attaching her feet together but we managed to catch her while she was eating under Mumu's wing and we removed them. In the meantime she divorced and found a more affectionate partner, TheTallOne, Mumu's rival!
Mumu was not a kind partner, he always pricked LittleWhite and did not let her eat, even after he had finished.
He is some kind of lonesome cowbow. He does not accept any other pigeon.

One day, shortly after New Year, this astonishing event: Mumu brought me a twig!!!
This was an incredibly emotional moment. And this happened several times since, at some periods.
One could say "he wants to mate with you" but my friend Mumu is not that stupid! He knows I am of another species and the "peanuts waitress"! He never pricked me and never cooed at me swelling his neck as he does with all pigeons! And anyway he had the same behavior with my husband

This is mysterious and overwhelming. He is showing me the twig, very excited and happy. A present? Or he wants to show me he is building a nest? (but no sign of a mate) He also plays with the twig and takes it away from me if I try to take it, like dogs do! Or he accepts a deal: a peanut or another twig against his twig!
Does someone here have observed something similar?

He is also very clever! I used to put a red triangle at my window as a sign to let him know I am not at home. I am not sure he understands exactly, but he certainly associates the red triangle with something negative.

If needed he accepts to eat in my hand, although I never tried to tame and cuddle him because I am worried about all what could happen to him and the other pigeons within the hostile environnement and people!

He used to stay a long time at my window edge, watching out for other birds and preventing them to approach, but he also seemed to enjoy simply our human presence and listen music.

I don't know what happened one day while I was not there, but he looked so scared for a while. Maybe a fight with TheTallOne? Or the neighbours chased him away? And as he had seen me feeding the other pigeons, he seemed very upset and had a strange behavior: he was shaking all over his body, had dilated pupils and did a kind of a catlike grunt, like a muffled cry (was it fear, anger, distress?) which was really thrilling. Since that time, we lost some closeness which makes me sometimes feel I made a mistake with permitting other birds to come.

I am wondering what his story is? Was he in close contact with humans before that could explain this singular behavior?

Every visit of Mumu is a sunshine! I feel happy and worried for him at the same time.
I could not imagine such a relationship with a feral bird, or any bird.
What a life lesson Mumu teached me! All these little souls in the city that nobody notices...

Here he is, Mumu, with his so beautiful and expressive eyes circled in white!