JUNEAU, Alaska » The night Lisa Murkowski announced she would mount a write-in campaign to retain her Senate seat, she acknowledged to a crowd of supporters that her odds were slim. Then she prompted a defiant roar: Invoking Native Alaskan culture, she told the crowd that the ancient Aleutian language contained no word for "impossible."

It was a deft play to the state’s strong sense of identity and a direct appeal to native communities, whose support could prove crucial. It was also inaccurate. The word in Aleut is haangina-lix.

"It’s very clear that you can say ‘impossible,"’ said Gary Holton, director of the Alaska Native Language Archive. "Clearly, she wasn’t checking her facts."

The facts line up starkly against Murkowski: The only person ever elected to the U.S. Senate as a write-in candidate was Strom Thurmond of South Carolina, in 1954. No one in Alaska has ever been elected statewide as a write-in. Joe Miller, the Tea Party star who narrowly defeated Murkowski in the Republican primary, now has the support of the Washington Republican establishment and is raising money quickly.

So why do plenty of people here, from analysts to many rank-and-file Republicans, think that Murkowski, 53, who first came to office through the easiest route imaginable (her father, then the governor, appointed her in 2002), could well pull off the impossible?

Because in a matter of weeks, she has morphed from establishment incumbent to renegade underdog. For many, it could be an ego-crushing transition, from sitting senator to plaintive write-in, but Murkowski is using it to her advantage, painting herself as the maverick in this race.

It was Sarah Palin who turned the state on its head in 2006 when she defeated Murkowski’s father, the incumbent, in a landslide in the Republican primary for governor. And of course there is Miller, a largely unknown lawyer who won last month’s primary by portraying Murkowski as a symbol of broken Washington and himself as the rebel.

Now Murkowski, already armed with a substantial campaign account and a well-known name (if not one that is easy to spell), has her own dark horse story.

"I failed as a candidate with my campaign in ensuring that Alaskans understood the urgency and why it was important that I retain this seat," the senator told a crowd of supporters who gathered here last week. "And further I failed in defending my record and quite honestly allowed it to be trashed there towards the end.

"I will not let you down again," Murkowski said to rising applause.

She has $1 million in her treasury, and her campaign hopes to raise another $1 million. Miller is trying to raise $1 million now.

Campaign events are set to begin in earnest this week, with a Tea Party event planned for next week.

Referring to the time between her write-in announcement Sept. 17 and the Nov. 2 election, Murkowski promised her supporters, "I’m going to give you the best 45 days of my life."

The crowd erupted. They had come to celebrate the opening of Murkowski’s campaign office here. The senator had not bothered to open one before the primary. But Juneau, Alaska’s capital, is one of the state’s strongest Democratic areas, and Murkowski needs Democrats, moderates and independents. Plenty seem ready to support her if it means defeating Miller.

Entering a cafe at the airport here alone, wearing a fleece jacket and with a backpack slung over one shoulder, the senator was soon surrounded.

"I’m so glad you’re doing this," said Patrick K. Gamble, the president of Alaska’s state university system, who was among those in the cafe.

Gamble, a former president of the Alaska Railroad and a retired Air Force general, emphasized that he was speaking of his personal views and not representing the university system, but he said the state had benefited from Murkowski.

"I admire leadership," he said. "I admire competence."

Alluding to the years since her initial appointment to office in 2002, he said, "She has earned her way."

Murkowski was elected to a full term in 2004, narrowly defeating Tony Knowles, a popular Democrat. She was aided then by a strong endorsement from Sen. Ted Stevens, her mentor in Washington. Last month, shortly before the primary, Stevens died in a plane crash. Now Murkowski is trying to claim his legacy while relying on a disparate group of supporters to help her wage essentially two campaigns.

One is more traditional: She argues that Miller is out of touch with Alaska’s needs in his conservative calls to cut federal spending and that the little-known Democratic candidate, Scott McAdams, is too inexperienced. The other campaign may be more difficult: getting a plurality of voters to write in her name on the ballot.

With five weeks to go, her campaign staff is being overhauled, with veterans of the 2004 race arriving in Alaska. Lawyers are also joining to help oversee what, if the race is close, could be a protracted and bitter vote count.

Murkowski said in an interview that the campaign is considering ideas like jingles and rubber bracelets to teach voters how to spell her name and fill in the oval beside "write-in" on the ballot.

The state elections director has said that any ballot from which voter intent can be discerned will be counted, but it is not thought that Lisa M. will suffice.

"This is what makes it so exciting, because there is no playbook," Murkowski said. "Nobody knows how to do this with certainty."

Her critics say Murkowski is a sore loser. Her supporters note that the 55,000 votes Miller received in the primary made up less than 12 percent of Alaska’s 490,000 total registered voters. Miller himself suggested he had work to do.

"Frankly, what needs to happen is we just have to get the message out to a broader audience," he said in a brief interview.

Asked how he would run against the revived Murkowski campaign, he said, "Same message, work harder." He noted that the senator had suggested that she would run aggressive ads against Miller, after barely engaging with him in the primary election.

"She’s announced her intent to sling mud, so we may end up having to have some reaction to that," Miller said. "Let’s hope not."

Miller benefited in the primary from a surge of conservative voters drawn to the polls by an abortion-related ballot measure. He was also endorsed by Palin and received more than $500,000 in support from the Tea Party Express. While there is no abortion-related measure on the ballot in November, Palin is expected to renew her support, as is the Tea Party Express.

"We’re going to bring a bunch of people back up there," said Amy Kremer, the group’s chairwoman, who lives in Atlanta. "I don’t think she can pull it off."

Murkowski’s resurrected candidacy could hurt the chances of McAdams, the Democrat, by stealing away votes. It could help him, too, by splitting the Republican vote.

"This thing," McAdams said in a brief interview, "is wide open."

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