TITLE: Are rounded rectangle billiard dynamics ergodic?
QUESTION [15 upvotes]: Bunimovich proved that the billiard-ball dynamics in the Bunimovich stadium is ergodic.
    
      
(Image from Microwave_billiards_and_quantum_chaos.)

Q. Is it known that the billiard-ball dynamics in a rounded rectangle is ergodic?

    
Here the corners are quarter circles; so can be viewed as a stadium with
vertical segment inserts. Perhaps Bunimovich's proof also covers rounded rectangles?
If anyone knows, I'd appreciate a pointer—Thanks!

REPLY [4 votes]: The relevant result is stated in Bunimovich, "Conditions of stochasticity of two-dimensional billiards" Chaos 1, 187-193 (1991): If the circles to which the arcs belong are completely contained in the billiard, the dynamics is ergodic.  So


Yes, if the rounded arcs have equal radius.


If not, it is easy to construct rectangles for which the above condition fails.


Conjecture: There is an example with non-equal radii which is not ergodic.