Almost all Collatz orbits attain almost bounded values
6 days ago
- #Dynamical systems
- #Collatz conjecture
- #Number theory
- The Collatz map, Col, is defined for natural numbers: Col(n) = 3n + 1 when n is odd, and Col(n) = n/2 when n is even.
- The Collatz conjecture posits that all orbits of the Collatz map eventually reach the value 1, a problem that remains unsolved.
- The research shows that almost all orbits eventually attain a value less than any function f(n) that tends to infinity, regardless of how slowly it grows.
- A key finding involves obtaining an approximately invariant (self-similar) measure for the accelerated Collatz dynamics.
- This work was presented as part of the Institute for Advanced Study's Members' colloquium.