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4:00 pm Monday, April 19, 2010 Ergodic Theory Seminar: Combinatorial classification of Markov partitions for Anosov and pseudo-Anosov automorphismsby
Alexey Klimenko (Steklov Mathematical Institute, Moscow) in HB 227- We consider a linear map x->Ax on a 2-torus, with A being a 2 x 2 hyperbolic matrix. It is well known that such system admits Markov parition, its elements being parallelograms with sides parallel to eigenspaces of A. These partitions can be obtained from a pre-Markov parition P (which is a bit weaker variant of definition of Markov partition) by taking connected components of intersections of element of P, an image of an element of P, ... and a (k-1)-st image of such an element. Pre-Markov partitions can have as few as 2 elements, so it is natural to make a description of ALL such partitions. The description given is the following: a) If we take any partition, there can be some others being its parallel translations. We give the algorithm to construct all these partitions and describe the number of such partitions. b) All partitions "up to translation" are naturally arranged in a bi-infinite sequence, the automorphism naturally acting on it by a shift S. In particular, the lift of such partition onto a plane can be one of two possible types. 1) The union of all lifts of one of partiton elements is a connected set, all lifts of another one are disjoint (``islands in the sea'' type) 2) Both unions of all lifts of a partition element are not connected (``herringbone parquet'' type) If the slope of a unstable eigenspace is decomposed into a continued fraction with a period (b_1,...,b_k), then the sequence of partitions described above is the following: there exists a fundamental segment for the shift S such that the partitions in it are: island, (b_1-1) parquets, island, (b_2-1) parquets, ..., island, (b_k-1) parquets. In particular, we see that the length of this segment is b_1+b_2+...+b_k. We also prove that for pseudo-Anosov homeomorphism of a surface there are just finite number of Markov partitions of the given complexity up to an action of the homeomorphism and give a (very inefficient) algorithm for enumerating them.
Submitted by damanik@rice.edu |