When President Biden and China’s leader, Xi Jinping, met on Saturday in Peru, they spoke directly to each other for perhaps the last time about a fierce superpower rivalry that Mr. Biden has sought to keep from spiraling into open conflict.

But both men also seemed to be addressing someone not in the room: Donald J. Trump, who has promised to take a more aggressive approach to Beijing when he becomes president again in January.

Mr. Xi, in his opening remarks, offered what appeared to be a stern warning as U.S.-China relations enter a new period of uncertainty after the American election.

“Make the wise choice,” he said in a conference hall at a hotel in Lima where the Chinese delegation was staying. “Keep exploring the right way for two major countries to get along well with each other.”

In his own opening comments, Mr. Biden seemed to try to make the case for maintaining a relationship with Beijing, as Mr. Trump talks about imposing more punishing tariffs on China and picks hard-liners for top administration posts.

“These conversations prevent miscalculations, and they ensure the competition between our two countries will not veer into conflict — be competition, not conflict,” he said.

“That’s our responsibility, and over the last four years I think we’ve proven it’s possible to have this relationship,” Mr. Biden added before the meeting, which lasted one hour and 40 minutes.

But even as Mr. Biden’s session with Mr. Xi, during a gathering of the Asia-Pacific Economic Cooperation forum, began with conciliatory words, it also gave the president a final chance to challenge the Chinese leader directly on some of the many issues that divide the two countries.

As Mr. Biden — and his vision for the world — heads for the exits, China’s recent actions suggest that it has little interest in placating Washington.

American officials have expressed alarm over China’s increasingly close relationship with the Russian president, Vladimir V. Putin. And Chinese hackers were recently accused of breaking into the American telecommunications system and obtaining information from the phones of U.S. officials.

Beijing has also continued to flex its muscle in Asia, with U.S. officials voicing growing concern about China’s military exercises around Taiwan and its broader aggression in the contested South China Sea.

Mr. Biden pushed Mr. Xi to maintain peace in Taiwan, and pressed the Chinese leader over Beijing’s support for Russia, according to his national security adviser, Jake Sullivan. Mr. Biden also urged Mr. Xi to discourage North Korea from continuing to support Russia in its war in Ukraine.

“He also pointed out that the P.R.C. does have influence and capacity and should use it to try to prevent further escalation and further expansion of the conflict” in Ukraine, Mr. Sullivan said, using the acronym for the People’s Republic of China.

In addition, Mr. Biden raised the U.S. goal of transforming a Kenya-led and largely U.S.-funded security mission in Haiti to combat gangs into a formal U.N. peacekeeping mission, which would provide more money and personnel. China has expressed opposition to such a shift. Mr. Sullivan said Mr. Xi did not indicate any change in his position.

It was unclear how receptive Mr. Xi would be to any of Mr. Biden’s entreaties, with the clock quickly winding down on his presidency. China is now focused on preparing for the return of Mr. Trump, who has threatened to punish Beijing with 60 percent tariffs on its exported goods. Mr. Trump has also chosen Senator Marco Rubio, Republican of Florida, who has taken a hard line on China, as his nominee for secretary of state.

In his remarks, Mr. Xi signaled an openness to cooperation with the incoming Trump White House.

“China is ready to work with the new U.S. administration to maintain communication, expand cooperation and manage differences,” he said.

Mr. Sullivan also acknowledged the difficulty of ensuring that the new administration would respect any commitment agreed to on Saturday, including an agreement by both nations to maintain control by humans, rather than artificial intelligence, over nuclear weapons.

“The incoming administration is not in the business of providing us assurances about anything,” Mr. Sullivan said. “They’ll make their own decisions as they go forward.”

The Biden administration in many ways continued the tough approach toward Beijing initiated by the first Trump administration, including on tariffs, while adding restrictions intended to slow the Chinese military’s technological advancements.

Mr. Biden’s top aides said that countering Beijing, particularly around the Taiwan Strait, must be a continued priority for the incoming Trump administration.

For the past year, the Biden administration has tried to smooth tensions with China after the relationship hit a low point over a Chinese spy balloon that drifted over the United States and was ordered shot down by Mr. Biden. The two nations also tangled after Mr. Biden twice called Mr. Xi a “dictator” last year.

The White House has pointed to its shuttle diplomacy and efforts to open lines of communication as important steps in avoiding a spiral in relations.

Mr. Biden and Mr. Xi last met face to face in November 2023 in Northern California, where officials from both nations signaled a small improvement in the relationship. Since then, Mr. Biden’s aides have described steady progress in talks with Beijing and credited China’s effort to stem the flow of materials used to produce fentanyl.

Still, even as Mr. Biden has sought to steady relations, the fierce competition between the two countries was on vivid display during the APEC meeting in Lima.

China has courted Peru as part of a broader strategy to invest in smaller powers that feel neglected by the West. Earlier this week, Mr. Xi joined President Dina Boluarte of Peru to open a Chinese-funded port in a city 40 miles north of Lima.

Washington has warned that such projects could be used by the Chinese military in the future and potentially bring national security risks.

Mr. Biden has often cited his long and complicated relationship with Mr. Xi as evidence of his diplomatic skills. He often speaks of getting to know Mr. Xi when he was the vice president in the administration of President Barack Obama.

After returning to the White House from China in 2011, Mr. Biden told his aides that “I think we’ve got our hands full with this guy.”

But soon, the relationship with Mr. Xi will no longer be Mr. Biden’s responsibility to manage.

Seven months into the war in Gaza, Mohammed Said al-Halimy began documenting his daily routine in earnest.

Mr. al-Halimy, known by his friends and online as Medo, already had a teenager’s knack for capturing sunsets, songs and life’s milestones in short video snippets. That life was fractured after Israeli bombs fell on Gaza in response to the Hamas-led Oct. 7, 2023, attack, destroying his university and forcing him into a makeshift beach campsite.

As months of fighting ground into the summer, and his displacement became more entrenched, Mr. al-Halimy turned his phone camera to the surreal experience of everyday reality in dystopian circumstances.

“I wanted to show something positive, some resilience despite the daily suffering,” Mr. al-Halimy, 19, said in a July interview, adding that he hoped to capture an “unseen side of our lifestyle.”

Palestinians trapped in Gaza have been recording the war since it began, in often harrowing videos that have given a close-up view of the Israeli bombardment to millions of people worldwide. Many of their posts — raw, personal and at times graphic — went viral early in the conflict as traditional news media outlets struggled to get reporters into the blockaded enclave.

Now, young Gazans are sharing a different window into their lives: their routines amid a year-old war with seemingly no end in sight.

Mr. al-Halimy began posting about the hourslong wait to fill containers with drinking water, about concocting recipes with limited food supplies, and about a new garden plot he created in the soil beside the tent encampment that had for months been his family’s shelter. Showing his new baby mint plant to his Instagram followers, he asked, “Tell me in the comments, what should I name her?”

More than 6,000 miles away in central Florida, Sierra Taft, 36, was watching, checking Mr. al-Halimy’s accounts regularly for updates and worrying about his well-being.

“He felt like somebody that if I had met face to face, I could be best friends with,” she said.

Life in Gaza Through Instagram

Some Palestinians in Gaza document how they cook meals over open fires, using whatever few ingredients are available. Others unpack aid boxes or share exercise routines where doorways double as pull-up bars. And some show how friendly football and chess games are squeezed between piles of rubble and long lines for water.

With a command of English and growing followings, these Palestinian creators share their perspectives and appeal for help using the language of online influencers around the world who have amassed vast audiences by filming the minutiae of their lives.

So when Palestinian creators like Mr. al-Halimy portray normal activities like exercising or cooking against the backdrop of war, it is “a language that reaches,” said Laura Cervi, an associate professor in journalism at the Autonomous University of Barcelona who has studied Palestinian activism online.

“It’s not a number. It’s not like the complex journalistic vernacular,” she said, adding that from the perspective of viewers, “It’s a guy like me that is telling me that he exists — in the way I exist.”

Before the war, Mohammed Faris said his favorite place was the gym. Mr. Faris, a Khan Younis resident, had just started his first year at Al-Aqsa University when the war broke out. His parents, employees of UNRWA, the main United Nations agency for Palestinian refugees, encouraged him to start documenting his life. Since April, he has been sharing his diet and exercise routines under the account “Gymrat in Gaza,” which has gained more than 180,000 followers.

“Why not grab this opportunity to talk to the world?” he said in a recent interview from Khan Younis while refilling his supply of water. Mr. Faris said he had raised nearly $13,000 online since he started posting videos, and hoped to eventually evacuate his family from Gaza. He said his audience enjoyed it when he incorporated memes and jokes. “I like to add this touch of sense of humor,” he said.

But he struggles with the instability of being displaced from his home and the scarcity of healthy foods, he said. Finding stable internet connections can be a challenge, and he sometimes waits hours for a video to upload.

“What I want people to receive from my vlogs is that we are trying to cope with the situation,” he said.

The fighting has pushed most Gazans into shrinking areas designated by Israel as “humanitarian zones,” though U.N. officials and aid groups have said that no place in Gaza is safe and they fear famine.

Some viewers have criticized Mr. al-Halimy and others like him, accusing them of sharing misinformation, or questioning their struggle given their lighthearted messaging.

“I’m just showing you the 1 percent of my life — the 1 percent that I’m trying to have fun,” he said in a video posted in May. He added: “We’ve been through hell.”

A Global Reach

Even before the war, young Palestinians were adopting the lighter tone of online social media to conduct what Dr. Cervi calls “playful activism,” pointing to TikTok trends that incorporate humor to make political points.

The insistence on sharing everyday routines or incorporating a lighter tone into material about the war, she added, is its own form of defiance.

“It’s very political because they’re saying, ‘We’re surviving and we will keep on,’” she said. Framing these videos as lifestyle content, she said, makes it more likely that social media algorithms will share them with a broader audience.

Activists elsewhere have used playfully framed videos on social media platforms to share messaging about other causes, Dr. Cervi said, such as the struggle to combat the killing of women in Latin America.

Researchers say that social media postings not only can elevate causes, but also tend to simplify them by removing nuance and centering each creator’s perspective. What seem like candid moments can actually be carefully chosen and edited for effect.

Mr. al-Halimy said creating videos helped him endure his everyday hardships.

“I do my best to set up new, bright sides of my tent life and make it a day to remember,” he said in an interview in the summer. “A moment of pain, to a moment of hope.”

A graduate of a high school for gifted students, Mr. al-Halimy had studied in Texas under a State Department program. He said in July that his family had decided to stay together in Gaza, instead of being separated. His online following was growing fast, and he hoped to raise enough money for them all to leave.

On Aug. 25, he shared his final video on Instagram. The next afternoon, according to a friend who was with him, Mr. al-Halimy was at a makeshift cafe in Khan Younis when he was struck in the head by shrapnel from an Israeli airstrike. His brother, Zeid al-Halimy, said that he died at a Khan Younis hospital.

The Israeli military said it was not aware of an airstrike in the area that day.

In the months since Mr. al-Halimy’s death, his followers have been rewatching his videos and have left dozens of tributes in the comments. Some vowed to plant mint in their own gardens to remember him, and a fund-raising effort for his family has surged to more than $137,000.

Weeks after his death, Ms. Taft, who had never met Mr. al-Halimy in person, said she still thought about him every day. She compared losing him to another recent blow, the death of a close school friend.

“It’s the same feeling of loss,” she said.

Other Palestinians she followed online are never far from her mind.

“I’m wondering who the next one is going to be,” she said.

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 TV Rain 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.

Vladimir Shklyarov, one of the world’s premier male ballet dancers, died on Saturday night, according to the Mariinsky Theater in St. Petersburg, where he was a principal dancer.

The theater did not say in its announcement how or where Mr. Shklyarov, 39, died.

Russia’s Investigative Committee, a federal law enforcement agency, began an investigation into Mr. Shklyarov’s death, according to RIA Novosti, a state-run news agency. “A preliminary cause of death was an accident,” RIA Novosti quoted the Investigative Committee’s office in St. Petersburg as saying.

Over a two-decade career, Mr. Shklyarov gained international acclaim, performing at the Metropolitan Opera in New York, the Royal Opera House in London and other prestigious theaters around the world.

Born in Leningrad, Mr. Shkylarov graduated in 2003 from the Vaganova Ballet Academy, a famed institution with nearly 300 years of history. Its graduates include Natalia Makarova and Mikhail Baryshnikov.

Mr. Shklyarov joined the Mariinsky directly out of school, and went on to perform in both contemporary and classical productions, including “Swan Lake,” “Don Quixote” and “Giselle.” He received multiple awards for his work, including the prestigious Léonide Massine International Prize in 2008. But the Mariinsky said that “the most precious for Vladimir” was receiving the title of an honored artist of Russia in 2020.

Information about survivors was not immediately available. In 2012, Mr. Shklyarov married another dancer, Maria Shirinkina, who also performed with the Mariinsky, according to Russian news media.

In a review of a performance during the company’s 2016 tour of the United States, a New York Times critic said, “The best moment came from Maria Shirinkina, supernaturally airborne, and Mr. Shklyarov, elegant and mournful, in a cobbled together extract from ‘Giselle.’ For a little while, they suggested another world.”

A former ballerina, Irina Bartnovskaya, wrote on Telegram that Mr. Shklyarov had been at home, on painkillers and preparing for foot surgery before his death. She said that he went out to smoke onto “a very narrow balcony” and fell five stories in “a stupid, unbearable accident.”

Her account could not be verified, but it echoed similar reports in Russian news outlets, including Fontanka, which quoted a Mariinsky Theater spokeswoman, Anna Kasatkina, as saying that he was on painkillers and expected to have surgery.

Around the gaping shaft of a disused mine, the South African police on Saturday waited to pounce on anyone trying to escape to the surface.

Their siege has gone on for weeks, trying to flush out hundreds of men accused of illegal mining in an abandoned gold mine, where they are now camped out near Stilfontein, in South Africa’s North West Province. Officers have cut off the miners’ supply of water and food, guarded every known exit and pulled up or cut ropes used to ferry supplies underground, according to images distributed by the police.

Now, they wait for the bedraggled miners, who some South Africans fear are dying of hunger and thirst, to turn themselves in.

“We are not sending help to criminals. We are going to smoke them out,” a minister in the president’s office, Khumbudzo Ntshavheni, said in a news conference on Wednesday. “Criminals are not to be helped, they are to be persecuted.”

The siege tactics, part of national crackdown on illegal mining, have ignited a debate in South Africa about upholding human rights protected by the Constitution and efforts to fight crime in a country with high rates of lawlessness. It has also renewed attention to mining’s dark legacy in South Africa, and the common but dangerous practice of illegal mining, driven by soaring unemployment.

Two more miners resurfaced on Saturday, joining others who have trickled out in recent days, a police spokeswoman, Brig. Athlenda Mathe, said. The police say that as many as 400 miners could still be underground, adding that some are also believed to be armed. The authorities have said they will not risk officers’ lives by sending them down the warren of tunnels.

The police did not respond to questions about how many people have emerged from the mine so far, who owns the mine, when the operation began or how many officers were involved.

The South African government has said it would not send aid to the miners, despite growing public concerns for their welfare. Nor have the police signaled any intent to change tactics.

“We are standing static,” Brigadier Mathe said by phone from the scene.

The South African Human Rights Commission, an independent body established by the country’s post-apartheid Constitution, said on Friday that it was investigating the police operation after it received a complaint from a local community leader. While condemning criminality, the commission urged the police to adhere to the Constitution’s imperatives on human rights.

The siege is part of a national crackdown, called Vala Umgodi, meaning “Close the Hole” in the isiZulu language, launched last December to combat illegal mining across the country. Mining has for centuries been a part of South Africa’s economy, but as the industry shrank and mine owners abandoned unprofitable sites, unaffiliated miners — often called artisanal — began digging through what remained, without legal permits.

As unemployment rates have soared over the last decade, more out-of-work or untrained miners have gone underground. They remain there for weeks at a time, often setting up illegal electricity and water supply system. Many such miners come from poorer neighboring countries, and some are relatives of migrant laborers who had worked in South Africa, researchers said.

Artisanal mining has also become part of a global criminal network dealing in illicit precious metals, according to the Minerals Council of South Africa, an industry group. Criminal syndicates involved in mining, known in South Africa as Zama Zamas, often operate as armed gangs, ambushing mine security or engaging in shootouts with rivals, the council said. Violence and crime often seep into surrounding communities, creating a secondary market for food, alcohol and sex work, according to a study by the minerals council.

Poor mining communities, like the one where the police operation is underway, have also become reliant on illegal or artisanal mines. In the town of Stilfontein, where many out-of-work people have become artisanal miners in the North West Province, community leaders begged police to allow them to help the miners.

After signing agreements that the police would not be held liable if something went wrong, volunteers set up a system of ropes tied to large rocks around the open shaft entrance. They lowered food and water and lifted ailing miners. On Thursday, a decomposing body was brought to the surface, which the police said they would investigate.

“A person cannot risk his life, and go into those dangerous situations just to get bread and meat for his kids,” Johannes Qankase, a community leader, told the broadcaster Newzroom Afrika.

Stilfontein is one of many dying mining towns in South Africa, said David van Wyk, who researches the legacy of mining in South Africa at the Bench Marks Foundation, a nonprofit that tracks South Africa’s progress since apartheid ended.

There are over 6,000 abandoned mines in South Africa, and trying to arrest illegal miners is “like catching moles,” he said. Instead, the country should have regulated artisanal mining and ensured that companies followed the law when they withdrew from sites, he argued.

“This problem is not a crime problem, it is a business and economic problem,” Mr. van Wyk said.

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.

Eight people were killed and 17 others injured in a stabbing attack on Saturday night in the eastern Chinese city of Wuxi, the police said in a statement, in what appeared to be a second burst of deadly violence in China in less than a week.

A person wielding a knife attacked people around 6:30 p.m. at the Wuxi Vocational Institute of Arts and Technology, the police in Yixing, a district of Wuxi, near Shanghai, said in their statement. A suspect, described as a 21-year-old man who had graduated from the college this year, was arrested at the school, the police said.

The stabbing took place only days after a man killed at least 35 people by driving a vehicle into a crowd at a sports center in the southern city of Zhuhai, in the deadliest known violent attack in China in a decade. The authorities quickly moved to censor information and commentaries about the attack, which also injured at least 43 people. The driver stabbed himself, the police said, and was in a hospital in a coma.

On the Chinese social media platform Weibo, news of Saturday’s stabbing was shared widely, with thousands of comments by users. “How many cases have happened this week…” one wrote. “Again!!” read another post.

By early Sunday morning, censors appeared to be scrubbing discussion of the attack, in keeping with the Chinese government’s usual strategy after mass tragedies. In such situations, the government has suppressed voices of witnesses and survivors, minimized public showings of grief and promoted only official accounts and assurances.

Violent rampages targeting random civilians are rare in China, where police surveillance is ubiquitous, but there have been a series of attacks over the past year.

In September, a 10-year-old student was fatally stabbed by a Chinese man near a Japanese school in southern China. In June, two Americans were stabbed in northeast China, and a Chinese national who tried to intervene was knifed in the abdomen. In May, at least two people were killed and 21 others injured when an assailant went on a rampage with a knife at a hospital.

Alexandra Stevenson contributed reporting.

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 Ganghwa 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.