TAILIEUCHUNG - Đề tài " Stable ergodicity of certain linear automorphisms of the torus "

We find a class of ergodic linear automorphisms of TN that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when N = 4. As a corollary, we obtain the fact that all ergodic linear automorphism of TN are stably ergodic when N ≤ 5. 1. Introduction The purpose of this paper is to give sufficient conditions for a linear automorphism on the torus to be stably ergodic. By stable ergodicity we mean that any small perturbation remains ergodic. So, let a linear automorphism on the torus TN = RN /ZN be generated by a matrix A. | Annals of Mathematics Stable ergodicity of certain linear automorphisms of the torus By Federico Rodriguez Hertz Annals of Mathematics 162 2005 65 107 Stable ergodicity of certain linear automorphisms of the torus By Federico Rodríguez Hertz Abstract We find a class of ergodic linear automorphisms of TN that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when N 4. As a corollary we obtain the fact that all ergodic linear automorphism of TN are stably ergodic when N 5. 1. Introduction The purpose of this paper is to give sufficient conditions for a linear automorphism on the torus to be stably ergodic. By stable ergodicity we mean that any small perturbation remains ergodic. So let a linear automorphism on the torus TN RN ZN be generated by a matrix A e SL N Z in the canonical way. We shall denote also by A the induced linear automorphism. It is known after Halmos Ha that A is ergodic if and only if no root of unity is an eigenvalue of A. However it was Anosov An who provided the first examples of stably ergodic linear automorphisms. Indeed the so-called Anosov diffeomorphisms of which hyperbolic linear automorphisms are a particular case are both ergodic and C 1-open which gives rise to their stable ergodicity. Circa 1969 Pugh and Shub began studying stable ergodicity of diffeomor-phisms. They wondered for instance whether 0 0 0 -1 10 0 8 010 -6 0 0 1 8 was stably ergodic in T4. More generally Hirsh Pugh and Shub posed in HPS the following question Question 1. Is every ergodic linear automorphism of TN stably ergodic This work has been partially supported by IMPA CNPq. 66 FEDERICO RODRÍGUEZ HERTZ This paper gives a positive answer to this question under some restrictions. Let us introduce some notation to be more precise. We shall call A pseudoAnosov if it verifies the following conditions A is ergodic its characteristic polynomial PA is irreducible over the integers and PA cannot be written as a polynomial in tn for any n 2. There is

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