Li-Yorke sensitive minimal maps. (English) Zbl 1134.37307
Summary: Let \(Q\) be the Cantor middle third set and \(S\) the circle, and let \(\tau : Q \to Q\) be an adding machine (i.e. odometer). Let \(X = Q \times S\) be equipped with (a metric equivalent to) the Euclidean metric. We show that there are continuous triangular maps \(F_i: X \to X\), \(F_i: (x, y) \mapsto (\tau(x), g_i(x, y))\), \(i = 1, 2\), with the following properties.
(i) Both \((X, F_1)\) and \((X, F_2)\) are minimal systems, without weak mixing factors (i.e. neither of them is semiconjugate to a weak mixing system).
(ii) \((X, F_1)\) is spatio-temporally chaotic but not Li-Yorke sensitive.
(iii) \((X, F_2)\) is Li-Yorke sensitive.
This disproves conjectures of E. Akin and S. Kolyada [Nonlinearity 16, 1421–1433 (2003; Zbl 1045.37004)].
(i) Both \((X, F_1)\) and \((X, F_2)\) are minimal systems, without weak mixing factors (i.e. neither of them is semiconjugate to a weak mixing system).
(ii) \((X, F_1)\) is spatio-temporally chaotic but not Li-Yorke sensitive.
(iii) \((X, F_2)\) is Li-Yorke sensitive.
This disproves conjectures of E. Akin and S. Kolyada [Nonlinearity 16, 1421–1433 (2003; Zbl 1045.37004)].
MSC:
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 54H20 | Topological dynamics (MSC2010) |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
