President Biden and China’s top leader, Xi Jinping, have sparred for years about how the world should be ordered.

Mr. Biden, who has described Mr. Xi as a “dictator,” has said the preservation of democracy was the “defining challenge of our age.” Mr. Xi has accused the United States of being the “biggest source of chaos” in the world and warned against dangerous Western liberal ideas.

Now, as the two meet as world leaders for probably the last time in Peru on Saturday, it is Mr. Biden’s vision of the world that appears to be in retreat. The U.S. president is exiting the global stage with his stature diminished after Americans voted Donald J. Trump back into power.

Mr. Xi, on the other hand, remains China’s most powerful leader since Mao Zedong, unfettered by term limits and surrounded by loyalists. He has blamed China’s economic troubles on American “containment.” He has expanded Beijing’s influence worldwide, including in what the United States considers its own backyard — a point Mr. Xi drove home this week by inaugurating a $3.5 billion Chinese-funded deepwater port at the start of his visit to Peru.

The Biden administration says the president wants to use the meeting in Peru, on the sidelines of the Asia-Pacific Economic Cooperation forum, to challenge Mr. Xi on Chinese hacking, human rights violations and threats against Taiwan. Mr. Xi, who has bristled against being lectured by the West, is unlikely to pay much heed to Mr. Biden — and might see the contrast in their political standing as a vindication of his views.

“Part of Xi likely privately celebrated the defeat of the Democrats in the U.S. as showing the strength of the Chinese system,” said Steve Tsang, director of the SOAS China Institute in London. “He will probably just go through the motions in his bilateral with Biden. He will almost certainly not make any significant concession to Biden on anything.”

The future of Mr. Biden’s signature policy toward China, which centers on competing without straying into conflict and on rallying like-minded democracies to counter Beijing, is unclear. Mr. Xi is likely to reiterate that the world is big enough for both superpowers and that they need to get along, in a signal to President-elect Trump, who has a penchant for confrontation and has vowed to impose steep tariffs on China.

China knows that “after Trump takes office, it is very likely that many of the promises made by Biden, many of the policies adopted or implemented measures will be completely reversed,” said Xin Qiang, deputy director of the Center for American Studies at Fudan University in Shanghai. As power changes hands in the United States, the country’s foreign policy has become increasingly inconsistent, Mr. Xin said, like “sesame cakes being flipped” on a hot griddle.

And after Mr. Trump takes power, the relationship between the countries could become more volatile. His picks for top posts in his administration — including Representative Michael Waltz for national security adviser and Senator Marco Rubio for secretary of state — are known for having spent years thinking about how best to pressure Beijing and change its behavior.

“Throughout the first Trump term, the Chinese made fun of President Trump and American democracy as a joke,” said Yun Sun, the director of the China program at the Stimson Center in Washington. “This time they are taking him much more seriously.”

As competition between the two superpowers has intensified, raising concerns about a war or an economic crisis, Mr. Xi has sought to show that he is doing his part to maintain peace with the United States for the sake of global stability.

Though Mr. Biden adopted many of the Trump administration’s tough measures, he did so with the addition of intensive diplomacy. That is credited with helping prevent a rivalry — one that spans military power, technology, trade and space — from veering into conflict. But frictions continue to pile on: This week, the United States accused China of a vast hacking campaign targeting American telecommunications companies, a claim China rejected as disinformation.

Whether Mr. Trump will seek stable ties with Mr. Xi is a question that looms large as he prepares to take office. Tensions over China’s claims in the South China Sea and over Taiwan, as well as Beijing’s support for Russia, make relations between Beijing and Washington a minefield that could quickly plunge the world into a crisis.

For Mr. Xi, his trip to South America — he will also go to Brazil — is also a chance to assert that China is a force for stability as a counter to Mr. Trump’s unpredictability. And there might be nowhere more pointed for China to flex its geopolitical muscle than in South America, given the region’s proximity to the United States and its sphere of influence.

China has been aggressively courting Peru over the last decade, replacing the United States as the country’s top trading partner in 2014. China says Peru, which has a substantial ethnic Chinese population, is the second-biggest recipient of Chinese investment in Latin America. In July, a Chinese military delegation marched in Peru’s Independence Day parade for the first time, along with the South American delegations that traditionally take part.

“China cherishes its traditional friendship with Peru,” Mr. Xi said on Thursday as he met with President Dina Boluarte, noting that it was their third meeting in one year. He also waxed poetic about links between Inca and ancient Chinese civilizations.

Later, the two leaders ceremonially opened the Chinese-backed port in the city of Chancay, 40 miles north of Lima. The virtual event featured drone footage of the complex, a sprawling facility along the Pacific Coast outfitted with giant blue cranes bearing the name of China’s largest state-owned shipping company. Mr. Xi also visited the Callao air force base in Lima.

China’s growing coziness with Peru highlights China’s strategy of courting smaller powers, particularly those that feel neglected by the West.

But China’s influence in the country has drawn concern. Allegations of corruption have trailed several contracts won by Chinese companies. Washington has warned that the Chancay port could be used one day by Chinese warships to threaten the United States.

Last year, Gen. Laura J. Richardson of the U.S. Army, who is now retired, warned that China was “right under our nose” because of its investments in South America. Those include telecommunications networks, railroads and space infrastructure.

“They’re on the 20-yard line to our homeland,” she said.

Berry Wang contributed reporting from Hong Kong.

It was the largest East-West prisoner swap since the Cold War, with a Russian assassin and seven others returned to Moscow in August in exchange for 16 prisoners who had run afoul of President Vladimir V. Putin.

Among those released by Russia were four political prisoners and three people with ties to the country’s most prominent opposition figure, Aleksei A. Navalny, who died in prison in February. The deal seemed poised to breathe new life into a fractured movement that had struggled to exert influence in the aftermath of Mr. Navalny’s death.

But three months later, there are signs that the Russian opposition movement has never been more divided — or faced as steep a challenge in working to counteract Mr. Putin.

Infighting and accusations among competing anti-Putin groups threaten not only its political and financial viability, but also the legacy that Mr. Navalny worked hard to leave behind.

On Sunday, legions of those opposed to the Kremlin’s rule are expected to march in Berlin in the first big anti-Putin protests since the activists were released in August — a rally intended as a strong show of unity.

The election this month of Donald J. Trump to a second term as the U.S. president has further complicated the effort. Mr. Trump in the past has expressed admiration for Mr. Putin and suggested that he would end American support to Ukraine in its war against Russia. If Mr. Trump takes a benevolent stance toward Russia, it could further insulate Mr. Putin from criticism.

Mr. Navalny’s daughter, Dasha, recently worked for the presidential campaign of Vice President Kamala Harris, Mr. Trump’s opponent. The opposition figure’s widow, Yulia Navalnaya, met with both Ms. Harris and President Biden this year.

Still, Leonid Volkov, one of the most influential figures in the Navalny camp, this month sought to assuage fears that a Trump administration would be worse for the Russian opposition than the Democrats. “We have never taken any goodies, freebies or favors from the Democrats — and we don’t expect any from the Republican administration either,” he said on a live show on the Navalny team’s YouTube channel.

Beyond the uncertainties of a new administration in Washington, however, the larger issue is a bitter fracturing of an opposition movement that has never spoken with one voice.

In September, Mr. Navalny’s organization made a shocking accusation: that another Putin critic in exile had organized a brutal assault with a hammer on Mr. Volkov, Mr. Navalny’s former chief of staff, in Lithuania in March.

Many opposition-minded Russians were already frustrated with Mr. Navalny’s allies over the past few years, viewing them as imperious and insular. But the accusation opened floodgates of criticism among Russian activists living in the West, with some prominent figures accusing Mr. Navalny’s aides of trying to silence any voices that might compete with them for leadership of the opposition.

Two weeks after the September accusation, the opposition was further roiled when an anti-Putin campaigner, Maxim Katz, accused the Navalny team of receiving funds from people accused of fraud, and even elevating one to register the group’s legal entity in the United States.

In the aftermath of these accusations, the crises engulfing the pro-democracy opposition have tarnished the reputation of a group that many hoped would be a beacon of leadership for a revived anti-Putin opposition.

“You have squandered Navalny’s legacy,” Boris Zimin, a Russian businessman in exile who had been one of the Navalny group’s biggest donors, wrote in a Facebook post directed at Mr. Navalny’s exiled aides.

Aleksei A. Venediktov, the former editor of Ekho Moskvy, a popular radio station that was shut down by the government after Russia’s invasion of Ukraine, lamented the fissures in the opposition.

A representative for the Navalny opposition group declined to comment.

“This is a competition to show the West who the real leadership of the Russian opposition is,” said Mr. Venediktov, a former subject of investigations by the Navalny group’s anti-corruption unit, known as the FBK.

“They are fighting for control of the cage they are in, without understanding that there is a whole territory around it to focus on,” he said.

The Navalny group made its accusations in a video report by its investigations unit, which it posted on YouTube. It said that the hammer attack on Mr. Volkov — initially assumed to have been ordered by the Kremlin — had actually been orchestrated by Leonid Nevzlin, a former businessman in Russia and now a prominent democracy campaigner in exile.

The report said the Navalny team had become convinced of that scenario after seeing chats and hearing recorded conversations that it said implicated Mr. Nevzlin.

Mr. Nevzlin, 65, denied all of the accusations. “I have nothing to do with any attacks on people, in any form,” he wrote on X the day the report was published. “I am convinced that justice will confirm the absurdity and complete groundlessness of the accusations against me.”

For weeks, the hourlong video produced by the Navalny team was the primary topic on YouTube channels popular with opposition-minded Russians.

In its video, the Navalny group sought to link Mr. Nevzlin with Mikhail B. Khodorkovsky, the former Russian oil tycoon who spent a decade in prison. He is now a major financier of opposition activities from exile and considered by some to be a rival of the Navalny group.

There was no implication that Mr. Khodorkovsky had been involved in the alleged plot. But the Navalny team’s efforts to implicate him, some observers say, further indicated that the Russian opposition is competing within itself for influence and the right to represent anti-Putin Russians, rather than building a big tent movement.

The accusations by Mr. Katz further divided the anti-Putin forces.

Mr. Katz, who worked with the Navalny team more than a decade ago but who has recently been critical of it, published a two-hour video and accompanying text. It accused the group of accepting money from bankers that he said had defrauded customers of hundreds of millions of dollars, and knowingly providing the bankers with political cover in the West.

The Navalny team and its bankers deny all wrongdoing.

On Wednesday, Ms. Navalnaya, Mr. Navalny’s widow, told the Russian TV channel Dozhd that the foundation was “going through very, very tough times right now.”

She acknowledged that it was a mistake to engage with “dubious people,” but she also thanked the banker who registered the organization in the United States, saying he “helped in a difficult situation” and that she believed he had done this “completely selflessly.”

Many commentators say they are still looking for much-needed answers from the Navalny camp.

“FBK behaves as an example of political purity and honesty and sets standards for anti-corruption investigations and responses to them, but it turns out that it itself does not meet these standards,” Mikhail Fishman, a journalist with the exiled TV Rain, said in a weekly commentary show released on Nov. 3. “This answer raises doubts about the credibility of all other FBK investigations,” he added.

Even some of Mr. Navalny’s most ardent supporters and his organization expressed disappointment over the recent developments. Mr. Zimin, the large Navalny donor who runs a family philanthropic foundation focused on education and science, said, “All the enormous credit of trust, political capital — everything went down the drain.”

In May, Mr. Zimin said in an interview that he would stop giving money to Mr. Navalny’s organization because he had found himself “increasingly at odds with what the FBK was doing.”

Mr. Katz, who has a large following online and sustains himself in exile largely on small donations from his viewers, said he envisioned a future in which Russian opposition forces could unite. But he said his repeated requests to meet with the Navalny team had been refused.

Mr. Venediktov, the former radio station editor, said that while he felt Mr. Navalny was a talented politician who knew how to bridge disagreements and build coalitions, the people now carrying on his legacy are less able to find common ground with people who share their aim of overthrowing Mr. Putin.

“Every single decision they make seems to damage the anti-Putin position,” he said.

Nataliya Vasilyeva contributed reporting from Istanbul, and Milana Mazaeva from New York.

Loud, crackly noises that sounded like an ominous, giant gong being beaten again and again washed over this village on a recent night. On other nights, some residents described hearing wolves howling, metal grinding together or ghosts screaming as if out of a horror movie. Others said they heard the sound of incoming artillery, or even a furious monkey pounding on a broken piano.

Although they heard different sounds at different times, people in this South Korean village on the border with North Korea all call themselves victims of “noise bombing,” saying they find the relentless barrage exhausting.

“It is driving us crazy,” said An Mi-hee, 37. “You can’t sleep at night.”

Since July, North Korea has amped up loudspeakers along its border with South Korea for 10 to 24 hours a day, broadcasting eerie noises that have aggravated South Korean villagers like no past propaganda broadcasts from the North ever did. The offensive is one of the most bizarre — and unbearable — consequences of deteriorating inter-Korean relations that have sunk to their lowest level in years under the North’s leader, Kim Jong-un, and the South’s president, Yoon Suk Yeol.

For decades, the two Koreas — which never signed a peace treaty after the 1950-53 Korean War ended in a truce — have swung between conciliatory tones and saber rattling. Under Mr. Kim, Pyongyang has veered toward a more hawkish stance over the past few years. It has shut off all dialogue with Seoul and Washington, doubled down on testing nuclear-capable missiles and has vowed to treat South Korea not as a partner for reunification, but as an enemy that the North must annex should war break out.

In the South, Mr. Yoon has also adopted a more confrontational approach since taking office in 2022. He has called for spreading the idea of freedom to the North to penetrate the information blackout Mr. Kim relies on to maintain his totalitarian rule. South Korea has also expanded joint military drills with the United States and Japan, which involved aircraft carriers, strategic bombers and stealth jets, to deter Mr. Kim.

Complicating the global picture, North Korea this year strengthened its ties with Moscow, shipping weapons and troops to aid its war against Ukraine and striking a mutual defense pact in the event either is attacked.

The souring of ties is increasingly affecting the lives of people living along the Demilitarized Zone between North and South Korea, where Mr. Kim’s growing hostilities toward the South have taken the form of noise bombardment.

“It’s bombing without shells,” Ms. An said. As she spoke from her living room, the distant gong-like sounds outside raged on, the noise seeming to grow louder as the night deepened. “The worst part is that we don’t know when it will end, whether it will ever end.”

Ms. An’s village, Dangsan, has a population of 354, with most residents in their 60s and older. It has been one of the hardest hit by North Korea’s psychological warfare. Sitting on the northern shore of Gwanghwa Island, west of Seoul, it is only a mile from North Korea, separated by a stretch of gray sea.

“I wish they would just broadcast their old insults and propaganda songs,” said An Seon-hoe, 67, another villager. “At least they were human sounds and we could bear them.”

Since the 1960s, loudspeakers have been as much a fixture of the DMZ as razor-wire fences and land-mine warning signs. People living along the border endured propaganda broadcasts as a part of frontier life, as the rival governments switched them on and off, depending on the political mood.

When they were on, both sides insulted each other’s leaders as “puppets.” A female voice that drifted across the 2.5-mile-wide DMZ beckoned South Korean soldiers to defect to “the people’s paradise” in the North. South Korean broadcasts tried to entice North Korean troops with sugary K-pop tunes.

The latest bombardment from the North contains no human sound or music — just nonstop noises that villagers find hard to describe, other than calling them “irritating” and “stressful.” They have blamed them for insomnia, headaches, and even goats miscarrying, hens laying fewer eggs and the sudden death of a pet dog.

The noise was part of a series of steps North Korea has taken to retaliate against what it called South Korean hostility. Recent events might explain why the sounds have become so intolerable.

Since his negotiations with President Donald J. Trump collapsed in 2019, Mr. Kim has shifted the course of his country’s external relations, turning increasingly hostile toward South Korea, in particular.

Some analysts say that by raising tensions, Mr. Kim was building the case for why the next American president needed to engage with him as he seeks an easing of international sanctions in return for agreeing to contain his nuclear program. The impending return of Mr. Trump, who is now president-elect and with whom Mr. Kim met three times during his first term, could increase the chances of the two countries’ engaging again after years of silence.

But others say Mr. Kim’s recent rhetoric toward the South reflected a fundamental shift, channeling his belief in the advent of a “neo-cold war.”

The catalyst for this change was waves of anti-Kim propaganda leaflets that were sent across the border via balloons by North Korean defectors living in the South, said Koh Yu-hwan, a former head of the Korea Institute for National Unification. These leaflets called Mr. Kim “a murderous dictator” or “pig” and urged North Koreans to overthrow his government.

In May, North Korea retaliated by sending its own balloons to the South, loaded with trash in response to what Pyongyang called political “filth” from the South.

Weeks later, South Korea ended a six-year hiatus in propaganda broadcasts, switching its loudspeakers back on to blast K-pop and news to the North. The North responded with its blasts of strange, nerve-racking noises.

“North Korea knows its propaganda no longer works on South Koreans,” said Kang Dong-wan, an expert on North Korea at Dong-A University in the South. “The goal of its loudspeakers has changed from spreading propaganda to forcing South Korea to stop its own broadcasts and leaflets.”

Until inter-Korean tensions caught up with them, Dangsan residents were proud of their quiet rural life despite their proximity to the border. They grew red peppers and thick radishes in their gardens. Cats sauntered under persimmon trees strung with heavy fruits. Wild geese took off from harvested rice fields in a chorus of honking.

These days, however, villagers keep their windows shut to minimize the noise from North Korea. Some have installed Styrofoam over them for extra insulation. Children no longer play on outdoor trampolines because of the noise.

Political leaders have visited Dangsan to offer their sympathies. During a parliamentary hearing last month, a teary Ms. An knelt before lawmakers, asking for a solution. But officials suggested neither a plan to de-escalate the psychological war with the North nor a solution to the noise, villagers said, other than offering double-pane windows for villagers and medication for their livestock to better endure the stress caused by the noise.

“The solution is for the two Koreas to recommit themselves to their old agreements not to slander each other,” said Mr. Koh, of the Korea Institute. But things have only worsened. Last month, North Korea demolished all railway and road links between the two Koreas with dynamite. This month, it disrupted GPS signals near the western border with the South, affecting some civilian ship and air traffic, according to the South Korean military.

Residents near the border have grown weary of ebbs and flows of tensions on the peninsula. Ms. An’s father, An Hyo-cheol, 67, the village chief of Dangsan, urged the South Korean government to stop what some villagers called a “childish” shoving match with the North. He demanded that the Yoon administration stop all propaganda broadcasts and ban leaflets, to encourage the North to follow suit.

Dangsan residents said they were being sacrificed in the uncompromising political rivalry between the two Koreas.

“The government has abandoned us because we are small in number and mostly old people,” said Park Hae-sook, 75, a villager. “I can’t imagine the government doing nothing if Seoul suffered the same noise attack as we have.”

Shortly after she spoke, the afternoon offensive started with faint metallic howls coming across the border.

Ten newborns were killed in a hospital fire in the Indian state of Uttar Pradesh, the latest in a string of similar tragedies that highlight the abysmal state of building safety and fire preparedness in the world’s most populous country.

The fire, which broke out late Friday at the Maharani Laxmibai Medical College and Hospital in Jhansi, was caused by an electrical short circuit on the ground floor, according to Sachin Mahur, the chief medical superintendent of the government-run facility.

That was the location of the neonatal intensive care unit. Flames quickly engulfed the ward, which held 49 infants, Mr. Mahur said, speaking via telephone from the hospital. All the victims were less than a year old and on life support. Of those saved from the fire, 17 remained at the hospital, while others went home to their parents or were moved to another hospital, he said. A nurse also suffered burns.

“Some of the newborns who died were under observation after their treatment was over and to be sent home in the next day or two,” Mr. Mahur said. But the fire spread so quickly that it was impossible to save them, he said.

Hospital fires are not uncommon in India. In May, seven newborn babies lost their lives in New Delhi when a fire broke out at the private neonatal clinic. In one of India’s worst such fires in 2011, 93 people died in a private hospital in Kolkata.

“Even though over a decade has elapsed since that disaster, no lessons seem to have been learnt because the frequency with which accidental fires keep breaking out in hospitals has not reduced,” wrote the authors of a 2023 study on fire accidents in India. The study found that despite new building codes and safety systems to prevent fires or reduce their severity, implementation remains lax.

Images of the charred ward in Jhansi and of distraught parents were shared on social media. “Who will return my baby?” one cried in front of TV cameras.

The government announced compensation of five hundred thousand rupees, or about $6,000, for those parents who had lost their babies.

India is woefully understaffed and underequipped to manage fire outbreaks. Data provided to the Parliament in 2019 showed that the country had only 3,377 fire stations when regulations called for 8,559. The fire service had about 55,000 people, when a half-million were called for, and 7,300 vehicles, when it should have had 33,000.