A new study from Pew Research offers unsurprising news that many countries have a low opinion of Mr Donald Trump. This survey of 37 countries found the percentage of those with confidence in the United States President has fallen from 64 per cent at the end of the Obama presidency to just 22 per cent under Mr Trump. Some 62 per cent say Mr Trump is "dangerous", and 74 per cent have "no confidence" in him. Fewer than one in three support his bid to block citizens of some majority-Muslim countries from entering the US. Fewer than one in five approve of his trade and climate policies.

The fall is steepest among some close US allies. From 2015 to this year, the percentage of those with "confidence in the US President to do the right thing regarding world affairs" fell from 66 per cent to 24 per cent in Japan, 76 per cent to 22 per cent in Canada, 83 per cent to 14 per cent in France, and 73 per cent to 11 per cent in Germany.

Yet, officials in other countries know they can't simply ignore or isolate Washington. The US is still the only country that can extend political, economic and military influence into every region of the world. There are still international problems and challenges that demand US cooperation, if not leadership.

The good news for those who want more from the US is its decentralised federal system. Much power lies with state governors and big-city mayors to enact and enforce laws that don't exist at the federal level, even when these laws conflict with the President's priorities.

The most dramatic conflict between the national Republican Party, which now controls the White House and both Houses of Congress, and local-level politicians is over immigration policy.

Mr Trump has tried, so far unsuccessfully, to impose a ban on immigration from several majority-Muslim countries. But years ago, many local governments established "sanctuary" status for illegal immigrants, and that process continues.

"Sanctuary cities" refuse to cooperate with federal immigration laws and bar local police from questioning an individual's immigration status. According to The Centre for Immigration Studies, a non-profit that advocates a more restrictive immigration policy, about 300 cities, counties or US states have some form of sanctuary policy. On same-sex marriage, legalisation of drugs, and even voting rights, laws vary considerably from state to state.

US states, even its largest cities, have real economic heft. California's economy is larger than France's or India's. Texas is larger than Canada or South Korea. New York State is larger than Russia or Mexico. The state of Georgia's economy is larger than Nigeria's - Africa's largest. Los Angeles is larger than Turkey, and Chicago is larger than Sweden.

There's a wide variety of attitudes across US states towards Mr Trump and public demand within some to establish independent foreign policies. Mayors and governors, particularly in states where Mr Trump is deeply unpopular, can score political points by defying him and pursuing their own agendas. They can also benefit their states and cities by attracting more investment, more foreign students and more tourism.

No issue better illustrates the power of US states to set their own agendas than climate change. Here is a "problem without borders" that can't be addressed without cooperation from the US, which remains the second-largest emitter of greenhouse gases after China.

A few days after Mr Trump withdrew US support from the Paris climate accord, China's President Xi Jinping welcomed California Governor Jerry Brown into the Great Hall of the People with the sort of pomp traditionally reserved for visiting heads of state. The two leaders then discussed climate policy. "California's leading, China's leading," Governor Brown declared during a news conference covered extensively by China's state-controlled media.

California has established a "cap-and-trade market" that allows companies to buy and sell allowances on greenhouse gas emissions, a policy that finds little support at the federal level. Mr Brown, who has promised to set ambitious emissions targets in California whatever Mr Trump says, then signed agreements on clean-energy technological development with local Chinese officials.

Canada, far more dependent on the US economy than China, is adopting a similar approach to courting local US officials. Early efforts to court Mr Trump stalled when the President began complaining about unfair Canadian trade practices. Without needlessly provoking the US President, officials in Prime Minister Justin Trudeau's government have since begun building on already close ties between Canadian provinces and US states.

In fact, the day after Mr Trump explained his decision to withdraw from the Paris climate accord with a reminder that he "was elected to represent the citizens of Pittsburgh, not Paris", Canada's transport minister held a meeting on climate change policy with the mayor of Pittsburgh. Ontario will soon join Quebec in a cap-and-trade partnership with California. Canada's federal government is also building relationships with officials in Florida, Texas, Michigan, New York and other states.

"The United States is bigger than the administration," said Canada's environment minister recently. She's right.

There's nothing new, of course, about other governments, particularly US allies, forging political and commercial relations with US states and cities. But the Trump administration's "America First", often rejectionist, approach to the rest of the world has given these ties new urgency.

The US President has considerable power, particularly on foreign policy. But more governments are discovering the potential benefits of using the decentralised structure of the US to get what they want. And they will find a growing number of US governors and mayors waiting to embrace them.

• Ian Bremmer is the president of Eurasia Group and author of Superpower: Three Choices For America's Role In The World.