The continued fraction expansion of a real number may be studied by considering a suitable discrete dynamical system of dimension two. In the special case where the number to be expanded is a quadratic irrational, that is a positive irrational root of a polynomial of degree two, more insight may be gained by considering a new dynamical system of dimension three, where the state vector stores the coefficients of the quadratic polynomials resulting from the expansion process. We show that a number of constants of motions can be derived and exploited to explore the attracting set of the solutions. Links with the solution to Pell’s equations are also investigated.
Continued Fractions as Dynamical Systems
IAVERNARO, Felice;
2012-01-01
Abstract
The continued fraction expansion of a real number may be studied by considering a suitable discrete dynamical system of dimension two. In the special case where the number to be expanded is a quadratic irrational, that is a positive irrational root of a polynomial of degree two, more insight may be gained by considering a new dynamical system of dimension three, where the state vector stores the coefficients of the quadratic polynomials resulting from the expansion process. We show that a number of constants of motions can be derived and exploited to explore the attracting set of the solutions. Links with the solution to Pell’s equations are also investigated.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
