Piecewise rotations: limit set for the non-bijective maps

Characterization of Bijective Discretized Rotations

A discretized rotation is the composition of an Euclidean rotation with the rounding operation. For 0 < α < π/4, we prove that the discretized rotation [rα] is bijective if and only if there exists a positive integer k such as {cosα, sinα} = { 2k + 1 2k2 + 2k + 1 , 2k + 2k 2k2 + 2k + 1 } The proof uses a particular subgroup of the torus (R/Z).

Multiplicative bijective maps on standard operator algebras

A Classification of Bijective Polygonal Piecewise Isometries

We aim to give a classification of euclidian bijective polygonal piecewise isometries with a finite number of compact polygonal atoms. We rely on a specific type of triangulation process which enables us to describe a notion of combinatorial type similar to its one-dimensional counterpart for interval exchange maps. Moreover, it is possible to handle all the possible piecewise isometries, given...

Renormalization and Α-limit Set for Expanding Lorenz Maps

We show that there is a bijection between the renormalizations and proper completely invariant closed sets of expanding Lorenz map, which enable us to distinguish periodic and non-periodic renormalizations. Based on the properties of periodic orbit of minimal period, the minimal completely invariant closed set is constructed. Topological characterizations of the renormalizations and α-limit set...

Hierarchy of piecewise non - linear maps with non - ergodicity behavior

We study the dynamics of hierarchy of piecewise maps generated by one-parameter families of trigonometric chaotic maps and one-parameter families of elliptic chaotic maps of cn and sn types, in detail. We calculate the Lyapunov exponent and Kolmogorov-Sinai entropy of the these maps with respect to control parameter. Non-ergodicity of these piecewise maps is proven analytically and investigated...