TY - JOUR

T1 - Theory of (a, b)-continued fraction transformations and applications

AU - Katok, Svetlana

AU - Ugarcovici, Llie

PY - 2010

Y1 - 2010

N2 - We study a two-parameter family of one-dimensional maps and the related (a, b)-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations. We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.

AB - We study a two-parameter family of one-dimensional maps and the related (a, b)-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations. We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.

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U2 - 10.3934/era.2010.17.20

DO - 10.3934/era.2010.17.20

M3 - Article

AN - SCOPUS:77952999234

SN - 1079-6762

VL - 17

SP - 20

EP - 33

JO - Electronic Research Announcements of the American Mathematical Society

JF - Electronic Research Announcements of the American Mathematical Society

ER -