We investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai-Ruelle-Bowen measures (SRB measures), where the almost Anosov diffeomorphism is a system which is hyperbolic everywhere except for one point. At the indifferent fixed point, the Jacobian matrix is an identity matrix. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and can be expressed by using coefficients of the third order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed points.We discuss the relationship between the existence of SRB measures and the differentia- bility of some almost Anosov diffeomorphisms near the indifferent fixed points in dimensions bigger than one. The eigenvalue of Jacobian matrix at the indifferent fixed point along the one-dimensional contraction subspace is less than one, while the other eigenvalues along the expansion subspaces are equal to one. As a consequence, there are twice-differentiable al- most Anosov diffeomorphisms that admit infinite SRB measures in two or three-dimensional spaces; there exist twice-differentiable almost Anosov diffeomorphisms with SRB measures in dimensions bigger than three. Further, we obtain the polynomial lower and upper bounds for the correlation functions of these almost Anosov maps that admit SRB measures.
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