On some rational piecewise linear rotations

Abstract

AbstractWe study the dynamics of the piecewise planar rotations Fλ(z)=λ(z−H(z)), with z∈C, H(z)=1 if Im(z)≥0, H(z)=−1 if Im(z)<0, and λ=eiα∈C, being α a rational multiple of π. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of Fλ, with a period ℓ, that depends on the connected component. Furthermore, Fλℓ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides.Keywords: Periodic pointspointwise periodic mapspiecewise linear mapsfractal tessellationsMathematics Subject Classifications: 37C2539A2337B10 AcknowledgmentsWe thank our colleague Roser Guardia for the indications regarding the scale factor of the critical set that we mentioned in Section 4.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThe first, second and fourth authors are supported by Ministry of Science and Innovation–State Research Agency of the Spanish Government through grants PID2019-104658GB-I00 (first and second authors) and MTM2017-86795-C3-1-P (fourth autor). They are also supported by the grant 2021-SGR-00113 from AGAUR. The second author is supported by grant Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M). The third author acknowledges the group research recognition 2021-SGR-01039 from AGAUR.