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On One-Dimensional Contracting Repellers of \(A\)-Endomorphisms of the 2-Torus

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Notes

  1. Given a differentiable map \(f\colon M^n\to M^n\), a point \(x\in M^n\) is said to be critical if the differential \(Df\) at this point is not surjective.

  2. Let \(X\) be a topological space. A set \(A\subset X\) is said to be massive in \(X\) if it can be represented as a countable intersection of open dense sets.

  3. A map \(f\colon X\to X\), where \(X\) is a topological space, is said to be topologically transitive if there exists a point \(x\in X\) whose positive semiorbit is dense in \(X\).

  4. A set \(U_\Lambda\) is called a closed neighborhood of \(\Lambda\) if \(U_\Lambda\) is closed and \(\Lambda\subset\operatorname{Int}U_\Lambda\).

  5. The notation \(f^{-1}(U_\Lambda)\) is used for the full preimage of the set \(U_\Lambda\), and \(f^{-k}(U_\Lambda)\) is a shorhand notation for \((f^k)^{-1}(U_\Lambda)\).

  6. By \(h_f^{-1}(x)\) we mean the full preimage of the point \(x\).

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Funding

This work was supported by the Russian Science Foundation under grant 22-11-00027, except for the work on Theorem 1, which was supported by Laboratory of Dynamical Systems and Applications NRU HSE, grant of the Ministry of Science and Higher Education of the RF, agreement no. 075-15-2022-1101.

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Authors and Affiliations

  1. National Research University Higher School of Economics in Nizhny Novgorod, Nizhny Novgorod, 603155, Russia

    V. Z. Grines & D. I. Mints

Authors
  1. V. Z. Grines
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  2. D. I. Mints
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Correspondence to V. Z. Grines.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 613–617 https://doi.org/10.4213/mzm13850.

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Grines, V.Z., Mints, D.I. On One-Dimensional Contracting Repellers of \(A\)-Endomorphisms of the 2-Torus. Math Notes 113, 593–597 (2023). https://doi.org/10.1134/S0001434623030318

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  • DOI: https://doi.org/10.1134/S0001434623030318

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