Abstract
In this work, we introduce the notion of weak quasigroups, which are quasigroup operations defined almost everywhere on some set. Then, we prove that the topological entropy and the ergodic period of an invertible expansive ergodically supported dynamical system \((X,T)\) with the shadowing property establish a sufficient criterion for the existence of quasigroup operations defined almost everywhere outside of universally null sets and for which \(T\) is an automorphism. Furthermore, we find a decomposition of the dynamics of \(T\) in terms of \(T\)-invariant weak topological subquasigroups.
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Notes
The standard definition of expansiveness states that \((X,T)\) is expansive if there exists \(\delta >0\) such that if \(x\ne y\), then \(d(T^n(\mathbf {x}),T^n(\mathbf {y}))>\delta \) for some \(n\in \mathbb {S}\). Note that the standard definition implies the definition of expansiveness that we are adopting here.
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Acknowledgments
This work was supported by National Council of Technological and Scientific Development-Brazil Grants 304813/2012-5 and 304457/2009-4. The author was partially supported by the project DIUC 207.013.030-1.0 (UDEC-Chile).
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Sobottka, M. Standard decomposition of expansive ergodically supported dynamics. Nonlinear Dyn 77, 1339–1347 (2014). https://doi.org/10.1007/s11071-014-1383-4
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DOI: https://doi.org/10.1007/s11071-014-1383-4