VIANA MAPS DRIVEN BY BENEDICKS- CARLESON MAPS

Abstract

IN THE RESEARCH AREA OF DYNAMICAL SYSTEMS WITH HYPERBOLIC BEHAVIORS, VIANA MAPS REFER TO A CLASS OF DYNAMICAL SYSTEMS NAMED AFTER MARCELO VIANA, WHICH ARE SKEW-PRODUCTS OF QUADRATIC MAPS DRIVEN BY EXPANDING MAPS. IN THIS THESIS, WE CONSIDER A FAMILY OF VIANA MAPS THAT ARE CONSTRUCTED BY COUPLING TWO QUADRATIC MAPS. WE ARE DEVOTED TO STUDYING THE MEASURABLE DYNAMICS OF THIS FAMILY, ESPECIALLY THE ABUNDANCE OF NON-UNIFORM HYPERBOLICITY IN THIS FAMILY. USING COMPLEX ANALYTIC TECHNIQUES, WE PROVE THAT, FOR ANY POLYNOMIAL COUPLING FUNCTION OF ODD DEGREE, WHEN THE PARAMETER PAIR OF THE TWO FACTOR QUADRATIC MAPS IS CHOSEN FROM A TWO-DIMENSIONAL POSITIVE MEASURE SET, THE ASSOCIATED VIANA MAP HAS TWO POSITIVE LYAPUNOV EXPONENTS AND ADMIT FINITELY MANY ERGODIC ABSOLUTELY CONTINUOUS INVARIANT PROBABILITY MEASURES.