If there is such a place as the sports shoe capital of the world, this little Bavarian milltown is it.

On one side of the tiny Aurach River that divides this 975-year-old village is the headquarters of Adidas, the biggest and most well-known manufacturer of sport and leisure shoes in the world.

Growling at Adidas from headquarters on the other side of town is the cat face of Puma, Herzogenaurach's other world-famous "Sportschuhfabriken."

To Americans who are buying increasing numbers of the Adidas shoes with the three diagonal stripes or Puma's single flowing line, the two firms must seem like "the Volkswagen of the sport-shoe world."

But these two firms are much more than ordinary rivals for the multibillion-dollar market that goes with encasing the feet of the famous or merely the fantasy-minded in everything from featherweight sprinter's cleats to jogging shoes that ease the pain of neighborhood cobblestones.

They are owned by different sides of Herzogenaurach's most prominent family, the Dasslers - split by a bitter feud almost 30 years ago that lingers to this day.

Ironically, the effects of that quarrel over the years probably has made both families millions. Competitive pride has undoubtedly been a factor that has kept both companies a step ahead of other rivals in the techniques of producing high-quality track and sports shoes.

It also, at one unhappy point in their corporate histories, boosted their unofficial standing to one-two in the messy and illegal business of paying off star amateur athletes to wear their shoes in the all-important battle for recognition and prestige.

In the end, however, it seems to have paid off. The two firms - Adidas in particular - are among the most successful and perceptive international entrepreneurs.

They basically sensed and then caught the wave of affluence and sports consciousness that slipped over much of the world's middleclass in the past decade. They invested huge sums - outfitting track stars as well as entire soccer leagues around the world with free shoes and uniforms, knowing full well that when the television camera zooms in on the triple stipe or leaf trademark of Adidas, millions of spectators would make the connection.

Last Week, Adidas announced that the first winner of the 1980 Moscow Olympics had already been determined. They were right.

The firm "won" the right to outfit, free of charge, the thousands of officials at the next games. Adidas did the same thing at Montreal last year, supplying 7,000 athletic suits and jackets to officials at a cost of $250,000. In Moscow, Adidas says, there will be twice as many officials. But it will be worth it.

Adidas also has far outstripped Puma in sensing the boom in sports clothing - other than just shoes - as leisure dress for the more casual and well-heeled around the world.

Not to be outdone, Puma has its eyes on the World Cup of soccer in Argentina next year, which will also be watched by millions on TV. Puma has for years had most of the Argentine players under contract.

Of the two firms, Adidas is much bigger and generally regarded as more innovative. Although neither publishes financial statistics, Horst Dassler, the 40-year-old son of the man who founded Adidas, says sales last year were close to $500 million, including far more than $100 million in sales in the U.S.

Puma spokesman Sepp Dittrich says that firm now does about $100 million in sales a year, with sales in the U.S. up by some 35 per cent in recent years.

Adidas now claims it turns out 180,000 pairs of shoes a day from factories in 17 countries. Puma turns out 40,000 from plants in seven countries. Each says they export to almost 150 countries.

The roots go back to 1920, when Adolph Dassler, the son of a weaver in a boot factory, began a family business making slippers. By 1924, he was making sports shoes and his brother, Rudolph, joined him in the firm.

Adolf was an amateur athlete, loved soccer, and was an inventive technician. He soon made a better boot than the traditional English soccer shoe and also introduced the revolutionary concept of confortable leather training shoes.

In 1936, a black American named Jesse Owens won four Olympic gold medals in nazi-dominated Berlin in Dassler track shoes weighing 157 grams. Today, Adidas makes an olympic spring shoe with sharkskin sole and tiny spikes that weighs 89 grams, or three ounces.

The feud between the two brothers that was to split the family for three decades began around World War II and has never been reliably explained. Some trace it to the fact that Rudolph got drafted and Adolf did not and that while both were allegedly Nazi Party members, Rudolph was imprisoned after the war for a year by the Americans as a Nazi sympathizer.

By 1948, the two had split. Adolf, now 76, formed Adidas out of his name and began with 47 employees. Rudolph, who died in 1974, moved across town to start Puma with 15 workers.

After that, it was mostly Adidas that skyrocketed and mostly on the strength of Adolf's technical skills.

In 1954, a West German soccer team won the World Cup on a rain-soaked field in Switzerland.They were wearing Adidas Adidas shoes that had the first interchangeable cleats. In a world which, with the exception of the U.S., is soccer crazy, Adidas was never again short of customers.

This week, West Germany's national team took the field against Frnace outfitted for the first time with another potentially revolutionary soccer shoe that replaces traditional leather tops with a nobby, rubberized surface Adidas technicians say should provide better ball control.

By the 1956 Olympics in Melbourne, 75 per cent of all track and field winners were wearing Adidas, said Dassler, who now runs most of the company's business. The same was true in Rome in 1960. Last year in Montreal, 83 per cent of all single-event track and field medals were won by Adidas wearers.

But a not-so-funny thing happened between those games. In Rome, just before the 1960 Olympics, a German track star, Armin Hary, was widely reported to have been approached by Puma and offered money to promote its shoes.

By 1964 in Tokyo, Dassler said in an interview at the family home here, "things really started."

The firm, he said, had never "started to buy any athletes before. But the competition, not just Puma, used additional means to get their shoes used.

"Things really got bad in 1968 in Mexico City, where a series of incidents created a bad climate. Puma began offering money left and right to wear their shoes," Dassler claimed. "Then we found out. A few athletes did change. Some came to us and asked what to do. Finally we had to do the same thing to maintain the same number of athletes."

No names or figures were mentioned at the time, although Sports Illustrated magazine in 1969 claimed the firms' mutual antagonism yielded some $100,000 in payoffs, a figure that both companies say is far too high.

One American track star, George Young, finally put it on the public record in his book that says he was offered $4,000 by Puma but decided to stick with Adidas after they matched the offer.

"It was a question of prestige," says Puma's Dittrich. "The result of the big fight between the brothers and the athletes were not stupid. They would run from one to the other. It was a silly game."

It got so bad, Dittrich joked, that a soccer player would want several times what a sprinter wanted because he played for 90 minutes while the trackman was off the screen in 10 seconds.

But since Mexico, Puma's Dittrich claimed, "All this money business is finished now. Absolutely."

"We don't offer money to amateur athletes at all anymore. Totally out," said Dassler, who added that the two firms even had an informal agreement in Munich in 1972 to call each other if the problem arose.

"Munich was relatively quiet," he said, but added that in Montreal there were still some payoffs by some unnamed competitors.

The two German firms, of course, were not alone in offering payoffs and, as Dassler points out, even governments are in the act, paying much more than declared travel expenses in the bid to win medals and boost national pride.

Amateur athletes, Dassler said, "should be entitled to get something out of what they do. How to do it, to make it all open, is the Olympic committee's problem. The rules should be modified to allow some kind of benefit to the athlete."

Dassler said amateur athletes need support year round and his firm, and others, do it here by helping to sponsor clubs and giving them free equipment.

From 1948 on, the brothers Adolf and Rudolph spoke to each other exclusively through lawyers, who kept up a steady barrage of legal actions.

Since Rudolph has died, some of the emotion has died away, too. Said one Puma executive. "If we happen to meet on a plane now, we might say hello. But we don't talk about shoes."

Puma officials said Rudolph's death may also make the firm less conservative under the new chief, 47-year-old Armin Dassler, Rudolph's son.

Although Puma also makes top-quality equipment, its coups have generally been in the field of gathering endorsements, convincing and paying people to switch from Adidas.

Walking through a factory in rural Bavaria, it seems strange to see pictures of Catfish Hunter, Fran Tarkington and Walt Frazier in their Pumas.

Both firms - especially Adidas - have made huge inroads into the U.S. sports market. "Four or five years ago," said Adidas' Dassler, "75 per cent of the NBA and ABA players were wearing our shoes. Now it is down to about 40-50 per cent."

Dassler rated the U.S. company Converse highly but claims Converse "didn't pay until we came in with better technology."

Paying the pros is of course legitimate, Dassler said. "But it's not technology alone now. It's money."

Dassler estimated it costs $3,000 to $10,000 a year just to get an American professional athlete to wear a specific shoe, depending on the status of the star. Endorsements and personal contracts run from $40,000 to $100,000, he estimated.

In this year's Super Bowl, half of the players wore Adidas.

For much of the world, however, soccer remains the key to shoe prestige. Fourteen of the 20 first-division German clubs are outfitted free by Adidas, the other six by Puma. All but one of the top 29 French teams, said Dassler, wear Adidas.

Although Germany and France still outrank the U.S. as Adidas customers, in the long run, Dassler acknowledged, it is the U.S. that will be the biggest market.

"We use the same principles to sell in America as we use everywhere. But American youth attitudes to sports are even more open than elsewhere. Americans are more sports and leisure-minded than others. They dress more leisurely than Europeans. We manufacture mainly for comfort and quality-minded and that's the key to success," Dasser said.

A walk through both factories reveals a bewildering array of almost 300 different sport shoes. It strains an observer's credibility that there really needs to be separate models for high jumpers, depending if they use the "flop" or the "straddle" approach to the bar. There are high jump shoes just for the takeoff foot, marathon shoes, cricket shoes, parachuting shoes, long jump shoes, short- and long-distance jogging shoes, "grippers" for synthetic turf, different shoes for javelin, hammer and discus throwers.

But behind all of this, the real target is the average American or European who can afford the comfort of top Argentinian cowhide or kangaroo skin, likes the flair of a track shoe, and will wear it, as Puma's Dittrich put it: "Either for jogging or to go to the theater."