A sign greets visitors here, at the southern end of the Cyclades islands: Welcome to Amorgos. Nobody Will Find You Here! It’s a promise of an island off the beaten track, where timeless traditions are still the daily routine. The most famous of its landmarks is the cliff-side Hozoviotissa Monastery, a narrow white stack of rooms five stories high, clinging to the rock face like a swallow’s nest. The picturesque Agia Anna beach made it into the movies when director Luc Besson portrayed it in his film The Big Blue. The hilltop church of Agia Ioannis near Vroutsi offers breathtaking views of white villages: Katapola, Chora and ancient Kastri, as well as the endless, mesmerizing sapphire sea.