Ergodicity in the linked twist maps with counter-oriented twists
摘要
We reduce the earlier known optimal twist parameter for which ergodicity is established in the linked twist map with two linear twists in opposite sense, in the most general setting. Further, here we obtain ergodicity with possibly only one-fold twists in either lobe, while earlier results only applied for twist parameters at least 2. Almost hyperbolicity is easily established for twist parameters greater than 2, while in the most general setting of the linked twist map with both twists of equal magnitude, ergodicity was earlier established in the most general setting by Przytycki for twist parameters greater than 4.15 with at least two-fold twists in each lobe. Here we reduce this optimal twist parameter to 3.47 in the general setting. These techniques can be effected to make further improvements when additional assumptions are made on the dimensions of the strips or when the linked twist map is modified in other natural ways.