Reminiscences of Henri Epstein

From left to right: Louis Michel, Oscar Lanford, Henri Epstein, and Marcel Berger.

I remember Henri Epstein as a fun and helpful colleague.

When Michael Herman was lecturing on his work on the conjugacy of diffeomorphisms of the circle to rotations, Henri and I often walked across the valley together to attend his talks. Even though Michael’s lectures were highly technical, Henri persisted, showing his incredible tenacity in rigorously understanding every aspect of a mathematical proof.

Let me now tell you about how Henri helped me with a mathematics problem related to period doubling universality.

Henri showed that there is a real-analytic solution to Feigenbaum’s functional equation:

$$g \circ g(\lambda x) + \lambda g(x) = 0,$$

where $g: [-1, 1] \rightarrow [-1, 1]$ is an even function such that $g(0) = 1.$

He observed that the solution to this functional equation had inverse branches for which there exist complex analytic extensions to the entire upper half-plane. He also created computer-generated images of the complex fixed point of renormalization.

These real-analytic functions with complex analytic extensions arose naturally in my own eight-year struggle with period doubling universality, and I thus named these functions “functions in the Epstein Class.”

There are still many open questions regarding one-hump mappings of the line, particularly about the power used to shape the hump. Numerical experiments by physicists suggest that there is a universal geometry for each real power greater than one, but this is unproven except for even powers. In the case of even powers, the proof relies on complex analytic extensions of the forward mapping.

The argument shows that the backward branches, for any real power, have real-analytic renormalization limits within the Epstein Class. I believe this fact will play an important role in any future proof of this true, yet unproven, theorem.

Let me finish with another story involving Henri, related to the final stage of my struggle with period doubling universality.

By 1989, I was two-thirds of the way through proving the universality of the differentiable structure on the critical orbit Cantor set, defined by the infinite period doubling cascade. However, I still needed the complex bounds on these complex analytic extensions, and they were simply not forthcoming.

Bill Thurston was a huge presence in geometry at the time, and in my desperation, I decided to imitate him by “drawing pictures and using visual thinking” in a last attempt to establish the necessary bounds.

Here’s how my reasoning went:

The square root of the upper half-plane is a quarter-plane. Translate this to the right, take the square root again, and you get something with two right angles derived from the quarter-plane. Repeat this process over and over again, and you’ll end up with a fascinating image—very much in the spirit of Thurston.

I drew this on the blackboard in the visitor’s room, downstairs from Henri’s office on the second floor.

Suddenly, I remembered that some time back, Henri had produced the computer-generated images of the complex fixed point of renormalization mentioned above—specifically, iterations of the backward branches.

I ran upstairs to Henri’s office, but no one was there. The door, though closed, wasn’t locked—quite common at IHES in those days. Through the window, I saw a shelf with his reprints. I charged in, found the relevant paper, and flipped to the page with the figure showing the complex fixed-point Henri had discovered with the computer.

To my amazement, the figure looked exactly like the one I had just drawn on the blackboard downstairs.

This gave us both a clear picture and a precise conjecture to prove about the necessary bounds.

It still took me months to finish the proof. The final step was completed in July 1990 at the Orsay pool with Wellington de Melo. I later presented the proof at the Orsay dynamics seminar, and within a few weeks, Marguerite Flexor wrote up a rigorous account of my lectures.

Thank you, Henri, and rest easy,

Dennis Sullivan
New York, October 2024