Thomas Picketty in his Capital in the Twenty-First Century seems to be arguing that if the real return on capital is greater than the real growth in the whole economy, then capital's share of the economy will inevitably rise.
I don't see how that follows. It would be true if all the owners of capital were all crazed misers that never spent any of the income from their capital and reinvested everything for the benefit of their future selves. But they aren't.
In fact, Picketty suggests a more sophisticated model. If the generations of a rentier dynasty love their offspring like they love themselves and care nothing about anyone else, they will save just enough to keep their income growing at the same rate as the general economy, and consume the rest.
However, this results in capital's share of the economy staying static rather than growing.
But! The wealthy don't always care only about their own heirs. Warren Buffet and Bill Gates are only the latest examples of philanthropists who concluded that there were some people who needed their money more than their children did. For an earlier example, see Andrew Carnegie.
Also, even when a benefactor decides to pass his estate entirely to his heirs, you have the issue of multiplying heirs. An estate of $1 billion could easily by divided among 10 heirs two generations later.
Also, dynastic fortunes can go downhill quickly if something bad happens to the founder's industry and the heirs can't diversify quickly. Looking at the gilded age robber barons, you see a lot of fortunes built on railroads, steel and coal. Not industries that have done well since then. But, for example, the Vanderbilts couldn't just dump all their shares in the New York Central to diversify: not only would this depress the value of their shares, but a controlling interest was worth more to them than a minority stake.
Tyler Cowen reviews Picketty here. He raises good points.