The elasticity of son's income with respect to father's income is about 0.5 in the United States.
Regarding this study, Mankiw says that that 0.5 estimate is roughly the correlation between father and son income, ...and that the fraction of variance of son's income explained by father's income (R-squared) is only 0.25 and is called the "heritability" of a characteristic.
...some of the heritability of income must come not from inequality of opportunity but from the genetic transmission of talent. ...it would be shocking if we did not find some significant heritability of income. And that would be true even if equality of opportunity were perfect.
Needless to say, this low income mobility observed in the United States should be (1) the result of the tax system(Americans face lower tax rate than do Europeans) and, in my conjecture, should have been (2) an Americans' motivation to get ahead and, what's more, one of the causes of the dynamism of the American society: people work harder if they can get rewarded larger and longer. If they know they should pay more tax as they earn more, they are less likely to work harder from the outset and thus are more likely to earn at least no more than they used to earn.
I am not sure whether the above data can tell us about the heritability of character or talent(it is clear that we should research more on it to make it sure), and its resource research says nothing about the difference of race, the places where they live, career and education.
At any rate, we should be careful of the income data that we can get from some academic or official researches when we talk about the recent income gap observed in several developed countries.