CONTINUED FRACTIONS: ERGODIC AND APPROXIMATION PROPERTIES / FRAÇÕES CONTÍNUAS: PROPRIEDADES ERGÓDICAS E DE APROXIMAÇÃO

AUTOR(ES)
DATA DE PUBLICAÇÃO

2006

RESUMO

We study the theory of continued fractions emphasizing the interaction between theory of numbers (expansion of numbers, diophantine approximations, best approximations) and ergodic theory. We study the Gauss transformation and construct its ergodic measure. Using the Birkhoff Ergodic Theorem we obtain results about the expansion in continued fractions of almost every real number in [0, 1). We obtain properties about the approximation of real numbers by rational ones, the frequency of digits in the expansion by continued fractions, etc. We also study the Bernoulli shift and its relation with the Gauss map. Finally, we calculate the entropy of such a transformation

ASSUNTO(S)

transformacao de gauss transitividade transitivity convergents quotients entropy gauss map convergentes entropia quocientes

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