Ti comes from the French word 'Petit' which means little. So translated into English, this drink is a Little Punch. This is a very refreshing drink that provides a more multifaceted experience than its simple ingredients suggest. Take your time drinking a Ti-Punch and enjoy the aromas.
You need to use an agricole rum. I use Neisson's Blanc for its wonderful depth and flavor. It's sugarcane aromas are complimented nicely by the lime juice and peel.
You really want to use a quality cane syrup, preferably from one of the French islands. These syrups are robust in flavor and often a bit darker in color. I am using one from Martinique.
The lime is cut from the side so as to get more peel. It also directs all the juice into the drink instead of up into your eye. When you squeeze, this peel releases some lime oil which greatly enhances the aromas of the drink. In the Caribbean and in Mexico, the limes are quite often a lot smaller than those in the United States. You can basically only make 2-3 slices out of a lime.
The ratio of syrup to lime juice is a personal choice. I like to limit the syrup to 1/4 teaspoon, and squeeze in a little more lime juice. So I go slightly to the tart side. Experiment and find which ratio suits you. The ratio can depend on the rum you use. I usually stir with the 1/4 teaspoon measure so as to get all the syrup into the drink.