Efforts to review 2020 ballots in Georgia and Arizona reflect the staying power of Donald Trump’s falsehoods, and Democrats fear that the findings could be twisted by Republicans.Georgia has already counted its 2020 presidential vote three times, with the same result: President Biden defeated Donald J. Trump narrowly yet decisively. But now portions of the vote will be inspected for a fourth time, after a judge ruled late last week that a group of voters must be allowed to view copies of all 147,000 absentee ballots cast in the state’s largest county.The move carries limited weight. The plaintiffs, led by a known conspiracy theorist, will have no access to the actual ballots, Georgia’s election results have already been certified after recounts and audits showed Mr. Biden as the winner with no evidence of fraud, and the review will have no bearing on the outcome.But the order from Judge Brian Amero of Henry County Superior Court was a victory for a watchdog group of plaintiffs that has said it is in search of instances of ballot fraud, parroting Mr. Trump’s election lies. Election officials in Fulton County, which contains most of Atlanta, worry that if such a review does occur there, it could cast further doubt on the state’s results and give Republican lawmakers ammunition to seek greater power over the administration of elections.“Where does it end? It’s like a never-ending circus, this big lie,” Robb Pitts, the Democratic chairman of the Fulton County Board of Commissioners, said in an interview on Monday. “When they were accusing Fulton County and me in particular, I listened and I said — I said to the president, his representatives and I said to the secretary of state: ‘If you have evidence of any wrongdoing, bring it to me. If you do not, put up or shut up.’ And I repeat that again today.”The ruling in Georgia, a state that for months has weathered attacks from Mr. Trump and his allies as they falsely claimed the election had been stolen, coincided with a widely criticized Republican-led recount of over two million ballots cast in Maricopa County, Ariz., the largest county in another state that stunned Republicans by tipping to Mr. Biden last year after decades of G.O.P. dominance in presidential elections.That recount, which was approved by the Arizona state government and funded privately, resumed on Monday despite wide and bipartisan denunciations of the effort as a political sham and growing evidence that it is powered by “Stop the Steal” allies of Mr. Trump’s.The Arizona Republic reported on Saturday that volunteers being recruited to help recount the Maricopa ballots were being vetted by an organization set up by Patrick M. Byrne, the former chief executive of the online retailer Overstock.com and a prominent purveyor of conspiracy theories that the 2020 election was stolen from Mr. Trump.On Monday, an independent nonprofit news outlet, azmirror.com, reported that the organization conducting the hand recount, Wake Technology Services, had been hired in December for an election audit in Pennsylvania by a nonprofit group run by Sidney Powell, a onetime member of Mr. Trump’s legal team and prominent purveyor of conspiracy theories about the election.Late Monday, Mr. Trump continued to rail against the election results, citing the Arizona recount and the Georgia court ruling. “More to follow,” he said in a statement issued by his office. The efforts to continue questioning the legitimacy of the election in two critical battleground states, nearly seven months after voting concluded, illustrate Mr. Trump’s hold over the Republican Party and the staying power of his false election claims. Even though Mr. Trump is not directly involved in the continued examinations of votes in Arizona and Georgia, his supporters’ widespread refusal to accept the reality of Mr. Biden’s victory has led fellow Republicans to find new and inventive ways to question and delegitimize the 2020 results.A recount of over two million ballots cast in Maricopa County, Ariz., the state’s largest, was paused this month and resumed on Monday.Courtney Pedroza for The New York TimesLeading the Georgia ballot review effort is Garland Favorito, a political gadfly in Georgia who has lingered on the conspiracy fringe of American politics for decades. In 2002, he published a book questioning the origin of the attacks of Sept. 11, 2001. He has also trafficked in unproven theories about the Kennedy assassination and, in 2014, he appeared in a video promoting the idea that the 14th Amendment was itself unconstitutional and argued that the federal government was therefore illegitimate and should be overthrown.In an interview, Mr. Favorito cited his “15 years” of experience as a self-styled elections investigator, saying he had been first motivated by Georgia’s purchase of new election machines that did not maintain paper-ballot records. He said that his concerns about the 2020 election stemmed in large part from affidavits filed by former election officials who claimed that they had handled ballots that appeared to be counterfeit because they were either not folded, appeared to be marked by a machine, or were printed on different stock. (There is no evidence of widespread use of counterfeit ballots.)Though Mr. Favorito refused to accept the findings of the recounts and audits already done in Georgia, he said he would be satisfied if, after inspecting the ballot copies, he and his team found no problems.“Once we find out the truth, if the results were correct, we can all go home and sleep at night knowing that it was right all along,” Mr. Favorito said.But he does not view leading Republicans in Georgia — some of whom, like former Senator Kelly Loeffler, have been vocally supportive of his efforts — as allies.“The Republican establishment hasn’t reached out, whatsoever,” he said, adding that he had not voted for Mr. Trump but for a third-party candidate. And the funding for the inspection, he said, would come from “patriots” making small-dollar donations. “We don’t have any big money.”The spread and repetition of false claims about the election follows familiar patterns for disinformation, which often occupies segmented corners of the internet and social media. Forces both algorithmic and organic will surface content — such as theories of election fraud based on grainy social media videos or anonymous allegations — for people who are inclined to agree with it.But what have further fueled Mr. Trump’s election claims, aside from his continued public pronouncements, are the many lawsuits filed by the former president and his allies after the 2020 election.“Even though all of the lawsuits got thrown out, the Trump campaign did file a whole bunch of baseless lawsuits, which adds a layer of legitimacy when you’re reading about a lawsuit that’s been filed versus some rumor, allegation or piece of content online,” said Lisa Kaplan, the founder of Alethea Group, a company that helps fight misinformation. “It ratchets it up a notch.”The Georgia effort could also yet extend beyond the Republican echo chamber in which the 2020 election is still being litigated. The state’s new election law ensures that the General Assembly, which is currently controlled by Republicans, has broad authority over counties through a restructured state election board. The board can, among other things, suspend county election officials.As Mr. Favorito did a victory lap on pro-Trump news outlets, he won praise from top Georgia Republicans. David J. Shafer, the pro-Trump chairman of the Georgia Republican Party, emailed fellow Republicans on Friday calling Judge Amero’s ruling “a very significant and encouraging development.”Ms. Loeffler also praised Mr. Favorito’s effort.“While there is a dire need to investigate a number of other well-documented issues, we must also inspect Fulton County’s absentee ballots to reassure Georgians that their voices are heard and their votes are counted,” she said.Even Brad Raffensperger, the Republican secretary of state in Georgia, signaled support for the inspection led by Mr. Favorito’s group.“Allowing this audit provides another layer of transparency and citizen engagement,” Mr. Raffensperger said in a statement on Friday.The support from Mr. Raffensperger, who is now running for re-election, surprised some political observers in Georgia. It was the secretary of state who stood up to the false claims of election fraud in Georgia espoused by Mr. Trump and who has highlighted the audits conducted by state government officials last year as definitive reaffirmations of the election results. His office also filed an amicus brief in the lawsuit, arguing that Mr. Favorito’s group should not be given physical ballots for security reasons, though Mr. Raffensperger took no stance on the case in his brief.“From day one, I have encouraged Georgians with concerns about the election in their counties to pursue those claims through legal avenues,” Mr. Raffensperger said in his statement.Michael Wines More