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End-symmetric continued fractions and quadratic congruences

Barry R. Smith (2015)

Acta Arithmetica

We show that for a fixed integer n ≠ ±2, the congruence x² + nx ± 1 ≡ 0 (mod α) has the solution β with 0 < β < α if and only if α/β has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number α/β > 1 in lowest terms has a symmetric continued fraction precisely when β² ≡ ±1(mod α ).

Episturmian morphisms and a Galois theorem on continued fractions

Jacques Justin (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w . Then when

Episturmian morphisms and a Galois theorem on continued fractions

Jacques Justin (2010)

RAIRO - Theoretical Informatics and Applications

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w. Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

Ergodicity of ℤ² extensions of irrational rotations

Yuqing Zhang (2011)

Studia Mathematica

Let = [0,1) be the additive group of real numbers modulo 1, α ∈ be an irrational number and t ∈ . We study ergodicity of skew product extensions T : × ℤ² → × ℤ², T ( x , s ₁ , s ₂ ) = ( x + α , s ₁ + 2 χ [ 0 , 1 / 2 ) ( x ) - 1 , s ₂ + 2 χ [ 0 , 1 / 2 ) ( x + t ) - 1 ) .

Currently displaying 461 – 480 of 1815