[Paper Review] Dynamics of piecewise translation maps

Sang Truong|arXiv (Cornell University)|Oct 15, 2016

Mathematical Dynamics and Fractals2 references3 citations

TL;DR

This paper introduces a generalized piecewise translation map on Euclidean space, focusing on finite-type dynamics and attractor behavior. It establishes conditions under which the attractor is of finite type and conjectures semi-continuity of the attractor with respect to translation vectors, particularly in the case of m = d+1 branches, while also exploring double rotation maps on the 2D torus with numerical evidence for finite-type behavior and attractor stability.

ABSTRACT

In this paper, we introduce a generalized piecewise translation map on the Euclidean space. We provide a special case when this map is always of finite type. For a finite type map in this case, we form conjectures on the semi-continuity of the attractor. Moreover, we also provide some conjectures and pictures on piecewise translations on a two dimensional torus.

Motivation & Objective

• To define and analyze a generalized piecewise translation map on compact subsets of R^d with m branches.
• To investigate conditions under which the map is of finite type, particularly when m = d+1.
• To study the semi-continuity of the attractor under perturbations of translation vectors.
• To explore double rotation maps on the 2D torus and their finite/infinite type classification via numerical simulations.
• To conjecture that finite-type double rotations on the torus form a full measure, open, and dense parameter set.

Proposed method

• Define a piecewise translation map F: X → X, where X is the space of compact subsets of Ω ⊂ R^d, using m translation vectors v_i on partitioned regions B_i.
• Formulate the attractor as A = ⋂_{n=1}^∞ F^n(Ω), with F^n(Ω) forming a nested, non-increasing sequence of compact sets.
• Establish that the attractor is invariant under F using a lemma on intersections of compact sets and image mappings.
• Prove that if the attractor has non-empty interior and the number of iterations needed to enter the interior is uniformly bounded, then the map is of finite type.
• Use numerical simulations to visualize attractors of piecewise translations on a 2D disk and double rotations on the 2D torus.
• Conjecture semi-continuity of the attractor under Hausdorff metric by analyzing parameter neighborhoods and image propagation.

Experimental results

Research questions

• RQ1 Under what conditions is a piecewise translation map of finite type, particularly when m = d+1?
• RQ2 Is the attractor of a finite-type piecewise translation map semi-continuous with respect to perturbations of the translation vectors?
• RQ3 Do double rotation maps on the 2D torus exhibit finite-type behavior for almost all parameter values?
• RQ4 Can the attractor of a double rotation on the torus be semi-continuous in the parameter space of translation vectors?
• RQ5 Is the attractor continuous (not just semi-continuous) with respect to parameters in the finite-type region?

Key findings

• The attractor A = ⋂_{n=1}^∞ F^n(Ω) is always non-empty and compact, and is invariant under the map F.
• If the number of iterations required for F^n(Ω) to enter the interior of A is uniformly bounded, then the map is of finite type.
• Numerical results show that piecewise translation maps on a 2D disk converge to an attractor in as few as 8 iterations, indicating finite-type behavior.
• For double rotations on the 2D torus, simulations suggest that finite-type maps are stable and attain their attractor after 500 iterations.
• In contrast, some double rotation maps show no convergence after 5000 iterations, suggesting possible infinite-type behavior.
• The authors conjecture that the set of finite-type parameters on the torus is open, dense, and of full measure, supporting generic finite-type dynamics.