Abstract
We consider a topologically mixing hyperbolic attractor \(\Lambda\subset M^{n}\) for a diffeomorphism \(f:M^{n}\to M^{n}\) of a compact orientable \(n\)-manifold \(M^{n}\), \(n>3\). Such an attractor \(\Lambda\) is called an Anosov torus provided the restriction \(f|_{\Lambda}\) is conjugate to Anosov algebraic automorphism of \(k\)-dimensional torus \(\mathbb{T}^{k}\). We prove that \(\Lambda\) is an Anosov torus for two cases: 1) \(\dim{\Lambda}=n-1\), \(\dim{W^{u}_{x}}=1\), \(x\in\Lambda\); 2) \(\dim\Lambda=k,\dim W^{u}_{x}=k-1,x\in\Lambda\), and \(\Lambda\) belongs to an \(f\)-invariant closed \(k\)-manifold, \(2\leqslant k\leqslant n\), topologically embedded in \(M^{n}\).
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Notes
Let \(f:N\to M\) be a \(C^{1}\)-mapping. A point \(\in N\)is called a regular point if a differential \(Df\) is reversible. A point \(y\in M\) is called a regular value if the whole preimage\(f^{-1}(y)\) consists of regular points. Let \(y\in M\) be a regular value and for each \(x\in f^{-1}(y)\) let \(\varepsilon_{x}=1\) if \(Df(x)\) preserves orientation and \(\varepsilon_{x}=-1\) if \(Df(x)\) reverses orientation. Then a degree of the mapping f in the point y is defined as \(\deg_{y}(f)=\sum\limits_{x\in f^{-1}(y)}\varepsilon_{x}\). \(\deg_{y}(f)\) does not depend on a choice of a regular value \(y\in M\), so the degree of the mapping \(f\) can be defined as \(\deg(f)=\deg_{y}(f)\) where \(y\) is an arbitrary regular value of \(f\).
DA-diffeomorphism is a cascade on a torus \(\mathbb{T}^{2}\), first described by S. Smale in [18], which has a one-dimensional hyperbolic attractor and a fixed source and is obtained from an Anosov diffeomorphism via “Smale surgery”.
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This work is supported by grant 22-11-00027, except Theorem 1, whose proof was supported by the Laboratory of Dynamical Systems and Applications NRU HSE, by the Ministry of Science and Higher Education of the Russian Federation (ag. 075-15-2022-1101).
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37D05
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Barinova, M.K., Grines, V.Z., Pochinka, O.V. et al. Hyperbolic Attractors Which are Anosov Tori. Regul. Chaot. Dyn. 29, 369–375 (2024). https://doi.org/10.1134/S1560354723540018
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DOI: https://doi.org/10.1134/S1560354723540018