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Statistical properties of maps with indifferent periodic points. (English) Zbl 0996.37030
The authors deal with the extension of certain results for the unit interval that concerns the rates of mixing for the Manneville-Pomeau transformation. More precisely, they present an axiomatic approach to the decay of correlations for maps of arbitrary dimension with indifferent periodic points.
They consider a piecewise \(C^1\)-invertible map \(T:X\to X\) for a bounded region \(X\subset \mathbb{R}^d\) that is assumed to be expanding \((|\det DT|>1)\) except for a periodic orbit where \(|\det DT^q(x_0)|=1\). Assuming certain regularity conditions the existence of an absolutely continuous probability measure that is strongly mixing.
They apply the obtained results to the well-known Manneville-Pomeau equation and the inhomogeneous Diophantine approximation algorithm.

MSC:
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37A25 Ergodicity, mixing, rates of mixing
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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