In a pitcher, combine the simple syrup, apple juice, rum, and sparkling water. To serve, fill a martini shaker with ice and add about 1 cup of the rum mixture. Shake well and pour into martini glasses. Garnish with apple slices and serve.
1 1/3 cups Ginger Simple Syrup, recipe follows
1 1/2 cups 100% apple juice, chilled
1 cup dark rum, chilled
1/2 cup club soda or sparkling water, chilled
1 green or red apple, unpeeled, cored and thinly sliced