New examples of Besicovitch transitive cylindrical cascades. (English. Russian original) Zbl 1408.37074
Sb. Math. 209, No. 9, 1257-1272 (2018); translation from Mat. Sb. 209, No. 9, 3-18 (2018).
Summary: New examples of transitive cylindrical cascades with discrete orbits (the Besicovitch property) are constructed. For each \(\gamma\in(0,1)\) there exists a cylindrical cascade over a rotation of the circle, with a \(\gamma\)-Hölder continuous function, that has the Besicovitch property; furthermore, the Hausdorff dimension of the set of points on the circle which have discrete orbits is at least \(1-\gamma\). This improves (by \(\epsilon\)) an earlier estimate. In addition, an example of a cascade with discrete orbits such that the corresponding function satisfies the Hölder condition with each exponent \(\gamma\in(0,1)\) is constructed.
MSC:
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
| 28A80 | Fractals |
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