טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentAmram Guy
SubjectContinued Fractions and Geodesic Flow in Positive
Characteristic
DepartmentDepartment of Mathematics
Supervisor Professor Uri Shapira
Full Thesis textFull thesis text - English Version


Abstract

We study the relationship between continued fraction expansions in L=Fp((t-1)), the field of formal Laurent series in variable t-1, and the geodesic flow on the quotient of the projective linear group G=PGL2(L) by the integer lattice Γ=PGL2(Fp[t]). We review basic properties of continued fraction expansions and the continued fraction map T(x) ={x-1}. We discuss the quotient of G by the maximal compact subgroup K=PGL2(O), where O is the closed unit interval in L. We visualize G/K as the infinite p1 regular tree and use this visualization to study the geodesic flow on Γ\G. We use this geometric viewpoint to establish a correlation between measures invariant under the geodesic flow on Γ\G and T invariant measures on the open unit interval. We use this measure correlation to give a dynamic proof of the ergodicity of the continued fraction map with respect to the additive Haar measure on the open unit interval. We further discuss certain geodesics exhibiting escape of mass and derive conclusions about continued fractions of certain algebraic elements in L.