Danell Leyva‘s list of goals for this year and next hangs on a wall in the gym where he practices. He’s used to exceeding expectations. (Joshua Prezant/FOR THE WASHINGTON POST)

Danell Leyva watched a gymnastics video in his parents’ Miami home when he was 3 years old and fell in love. Now 19,Leyva remembers pointing to the screen and telling his mother and stepfather, who had spent their childhoods in Cuba’s national gymnastics academy, “That is what I want to do.”

His mother recalls the moment, too; only she considered the plan utter foolishness. When her gymnastics career ended, she had coached in Havana, hand-picking the island nation’s best young athletes, those with perfect gymnastics bodies.

Her son, whom she sneaked into the United States as a baby because he was in ill health, would never have made the cut. He had flat feet, too-long arms, breathing problems. He could not figure out how to jump and looked funny when he ran.

“I didn’t see in him the talent,” Maria Alvarez said. “I know gymnastics is so hard, and he was not coordinated and a little fat. . . . You have to be fast, strong, flexible. You have to be quick for gymnastics. Those things are most important, most necessary. . . . I said, ‘He is going to have trouble.’ ”

She was right: He did have trouble. She was wrong, however, about everything else. Last month, Leyva won his first U.S. all-around title at the 2011 Visa Championships in St. Paul, Minn., topping two-time champion Jonathan Horton and positioning himself as a medal favorite not only for the Oct. 7-16 world championships in Tokyo, but also for next year’s Olympic Games in London.

When he secured the victory, Leyva grabbed his coach and stepfather, Yin Alvarez, in a joyous bear hug and hoisted him several feet into the air. That moment perfectly captured Leyva’s dogged pursuit of his youthful dream: Every step of the way, his surprising successes have swept his parents right off their feet.

“He’s going to be in the top three in the world championships and watch out — next Olympics, he maybe will win the next Olympics,” Yin Alvarez said. “He is growing and growing and growing, and absorbing everything.”

Japan’s Kohei Uchimura, the two-time defending world champion, will be favored to win a third straight world title in his home nation in October, but Leyva is imagining another upset.

“I want to win,” he said. “I understand the true difficulty of beating someone as good as him. . . . I do feel like it’s hard — but not impossible. . . . I feel like I can do a lot better.”

A long journey

Leyva’s career has blossomed since he failed to make the 2008 Olympic team, but his start in gymnastics actually came years before he was born. He has lived an American dream with decidedly Cuban roots. His parents passed on to him the heavy-handed, disciplined education they received in Cuba, and which they surely could not — or would not — have offered had they never left home.

Yin and Maria Alvarez, who married in 2001, won admission to Cuba’s national gymnastics academy when they were in elementary school. They became fast friends, living in dormitories and training, competing and eating together, separated from their families but landing opportunities to travel and compete that less gifted children couldn’t fathom.

Even so, Yin Alvarez always dreamed bigger than Cuba’s political system allowed. He used to boast that he would someday own his own gymnastics school. Laughter usually followed; the government owns everything in Cuba, he would be reminded. He aspired to build an Olympic champion, but he knew in Cuba he would teach a student for only a year or two, then hand him off to another coach.

Alvarez’s father, who had been active politically against Fidel Castro, had escaped to the United States in 1968. In 1991, Alvarez decided to follow. He sneaked away from a Cuban gymnastics performing troupe during a trip to Mexico City.

He put his clothes in a plastic bag and swam naked across the Rio Grande River. His most powerful recollection: The water was freezing.

He caught a flight to Miami, met his father at the airport, then set off to find a place to teach gymnastics.

Practical and sensible, Maria Alvarez did not share her future husband’s dreamy visions. She decided to leave Cuba only after determining that her son’s health depended on it. Danell Leyva, whom she had with a man who now resides in Spain, became ill at 5 months old.

His breathing problems left her sleepless and scared. Every few weeks, she rushed him to the hospital. He suffered from a host of allergies and asthma.

Her journey with Leyva and his older sister detoured in Nicaragua and Peru and took months. When she finally arrived to Miami, she reconnected with her old friend Yin Alvarez. By 1995, he had secured financial support to open his own gymnastics facility. Her son, on medication for the problems that would cease when he turned 12, would attend public school while they coached side-by-side.

Leyva, however, had other plans.

“I’ve always said that I want to win the Olympics and win the Olympic gold,” Leyva recalled.

And he wanted to do it in gymnastics.

Not only did Maria Alvarez see Leyva as physically challenged, she also worried about all of the dust — from large bins of resin — around a gymnastics training center. Yet Leyva persisted, and his parents relented, allowing him to begin classes.

During his earliest days, he said, he was awful. He sobbed after finishing in last place on the high bar in his first competition at age 4 or 5. But instead of quitting, he trained harder. Yin Alvarez said his son learned pommel-horse skills well at a younger age than many accomplished gymnasts, including himself. Maria Alvarez recalled Leyva’s excitement after a seminar with an elite track and field sprinter; her son brought the running technique home and practiced it endlessly.

“I would be very persistent,” Leyva said. “I would get skills so much later than everyone else, but I would never stop. I would always keep going, keep going.”

Training ‘was a great time’

Maria Alvarez pulled Leyva out of regular school midway through first grade, arranging a tutor for him so he could train longer hours with the more advanced students.

Long training sessions turned into all day at the gym. The family usually arrived by 9 a.m. and left at 9 p.m. Leyva speculated that he had logged more hours at Universal Gymnastics — which proclaims itself “Home of Future Olympians” in massive lettering on the warehouse-size walls — than his own bedroom. He did his schoolwork there, made art projects there and helped his parents maintain the facility.

And he loved every minute of it.

“I wouldn’t trade it for anything,” Leyva said. “I thought it was amazing. I thought it was a great time. The friends I grew up with were always here. They’re still here.”

“When he start[ed], he [got] so serious about it,” Maria Alvarez said. “I used that as punishment. When he behaved bad or did something, I would say, ‘You are not going to do gymnastics.’ ”

Leyva’s persistence soon offset his alleged lack of talent. His stepfather requires that all of his male students write their goals at the beginning of each season on poster board that is taped onto the cement-block walls of the 18,000-square-foot gym. Leyva has been checking off his ambitions, one by one. In 2009, he became national champion on the horizontal bar. A year later, he finished second all-around at the U.S. championships and claimed the gold on the parallel bars.

At the world championships in Tokyo, Leyva will try to unseat Uchimura, a star he isn’t expected to beat. Maria Alvarez learned a long time ago not to doubt her son and his coach.

“They have a dream,” Maria Alvarez said. “I don’t have the vision like they have. I’m more day by day, more realistic. The realistic people don’t dream. . . . I [now] realize people who dream big go far.”