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**1 - 2**of**2**### CRITICALLY FINITE MAPS AND ATTRACTORS ON P k

, 2005

"... Abstract. We first study the structure of the post-critical set of critically finite maps on P k and show that the Julia set for a k−critically finite map is the whole of P k. We then study a specific k−critically finite map and show that the only non-empty closed backward invariant subset is the wh ..."

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Abstract. We first study the structure of the post-critical set of critically finite maps on P k and show that the Julia set for a k−critically finite map is the whole of P k. We then study a specific k−critically finite map and show that the only non-empty closed backward invariant subset is the whole of P k. Based on this result, we give an example of a holomorphic map on P k that has a chaotic nonalgebraic attractor. Finally, we study the dynamics of the map on the attractor. In particular, we construct an invariant hyperbolic measure supported on the attractor, which is mixing and is the unique measure of maximal entropy for the map restricted to the attractor. 1.

### ATTRACTORS ON P k

, 2006

"... We show that special perturbations of a particular holomorphic map on P k give us examples of maps that possess chaotic nonalgebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we construct invariant hyperbolic measures supported on the attractors wi ..."

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We show that special perturbations of a particular holomorphic map on P k give us examples of maps that possess chaotic nonalgebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we construct invariant hyperbolic measures supported on the attractors with nice dynamical properties.