We study a 4-fold symmetric kicked-oscillator map with sawtooth kick function. For the values of the kick amplitude λ = 2 cos(2πp/q) with rational p/q, the dynamics is known to be pseudochaotic, with no stochastic web of non-zero Lebesgue measure. We show that this system can be represented as a piecewise affine map of the unit square —the so-called local map — driving a lattice map. We develop a framework for the study of long-time behaviour of the orbits, in the case in which the local map features exact scaling. We apply this method to several quadratic irrational values of λ, for which the local map possesses a full Legesgue measure of periodic orbits; these are promoted to either periodic orbits or accelerator modes of the kickedoscillator map. By constrast, the aperiodic orbits of the local map can generate various asymptotic behaviours. For some parameter values the orbits remain bounded, while others have excursions which grow logarithmically or as a power of the time. In the power-law case, we derive rigorous criteria for asymptotic scaling, governed by the largest eigenvalue of a recursion matrix. We illustrate the various behaviours by performing exact calculations with algebraic numbers; the hierarchical nature of the symbolic dynamics allows us to sample extremely long orbits with high efficiency, i.e., uniformly on a logarithmic time scale.

Abstract. We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for λ ∈ { ±1±√5, ± 2 √ 2, ± √ 3} that all integer sequences (ak)k∈Z satisfying 0 ≤ ak−1 + λak + ak+1 < 1 are periodic. 1. introduction In the past few decades, discontinuous piecewise affine maps have found considerable interest in the theory of dynamical systems. For an overview, we refer the reader to [1, 7, 12, 13, 17,

We study the dynamics of renormalization of a specific interval exchange transformation which features exact scaling (the cubic Arnoux-Yoccoz model). Using a symbolic space that describes both dynamics and scaling, we characterize the periodic points of the scaling map in terms of generalized decimal expansions, where the base is the reciprocal of a Pisot number and the digits are algebraic integers. The set of periodic points has a rich arithmetic and geometric structure: we establish rigorously some basic facts, and use extensive numerical experimentation to formulate a conjecture.

Approach to a rational rotation number in a piecewise isometric system