Let

The Parry-Daniels map T has an exceptional set which can be seen as a strange attractor for T. The density of the invariant measure is given. Some remarks on the exceptional set for the mixture of the Selmer algorithm and the fully subtractive algorithm are added. Mathematics Subject Classification (2000): 11K55, 28D99. Key words: Ergodic theory, invariant measures. Let x ðx0; x1; x2Þ2 ðRþÞ3 and let be a permutation of the indices such that x0 x1 x2. The Poincare map P is defined as Pðx0; x1; x2Þ ðx0; x1 x0; x2 x1Þ: We introduce 2 fx2ðRþÞ3: x0 þ x1 þ x2 1g: Then the Parry-Daniels map T: 2! 2 is defined as Tðx0; x1; x2Þ x0 x2 x1 x0 x2 x2 x1 x2 2f"; ð01Þ; ð02Þ; ð12Þ; ð012Þ; ð021Þg: We introduce the notation xðkÞ ðxðkÞ0; xðkÞ1; xðkÞ2 Þ: Pkx