HONG KONG — What started in June with peaceful rallies in opposition to contentious legislation has devolved into a steady stream of mayhem, with some protesters embracing violent behaviors in response to brutal police tactics.

The protesters and the Hong Kong government, backed by Beijing, appear to have intractable differences, and there have been few signs of either side backing down.

In the months since the demonstrations began, the unpopular bill, which would have allowed extraditions to mainland China, has been withdrawn, but the protesters’ demands have expanded to include increased democracy and an investigation of the police.

The standoff has taken on international importance. China has viewed the protests as a challenge to its fervent nationalism, while democracy supporters worldwide have cheered what they see as a poke in the eye of the autocratic Chinese government. It all comes amid a rancorous trade war between China and the United States, and some international businesses, including the N.B.A., have found themselves stuck in a political mess they wanted no part of.