A riddled basin is a “basin of attraction [with] the property that every point in the basin has pieces of another attractor’s basin arbitrarily nearby,” explains the review paper “Fractal structures in nonlinear dynamics” (2009), by J. Aguirre et al.
In other words, a riddled basin is a region of the phase space that can be thought of as a bulky “fat fractal” boundary between different attraction basins. Every neighborhood of a point in a riddled basin, no matter how small, contains points that will eventually reach different attractors. Therefore, no matter how accurate is the specification of the starting point, the attractor that the system will eventually reach is undetermined.
Riddled basins, which have been found in many dissipative systems, “show that totally deterministic systems might present in practice an absolute lack of predictability,” note Aguirre at al. See also the book “Transient Chaos: Complex Dynamics on Finite Time Scales” (2011), by Ying-Cheng Lai and Tamás Tél.
I suspect that the fractal depth of riddled basins might be widespread in real-world, dissipative dynamical systems, and perhaps be the rule rather than the exception. If so, chaotic evolution is really nondeterministic.
Nature “knows” the starting point of the system as an infinitely precise real number. But we can’t know the starting point with infinite precision, and any finitely precise starting point contains the possibility of different outcomes.
In view of this, does it even make sense to claim that classical physics is deterministic in principle (though nondeterministic in practice)? Or shouldn't we, instead, follow Born and accept that the inevitably finite precision of initial conditions makes classical (non-quantum) physics nondeterministic in principle?
“As a mathematical tool the concept of a real number represented by a nonterminating decimal fraction is exceptionally important and fruitful. As the measure of a physical quantity it is nonsense… concepts which correspond to no conceivable observation should be eliminated from physics… the determinism of classical physics turns out to be an illusion, created by overrating mathematico-logical concepts.” - Max Born's Nobel Prize lecture.