Fooling Rebound Automata

We separate the class of languages accepted by deterministic two-way one counter automata from the languages accepted by two-dimensional rebound automata. We also discuss the relationship of the classes to languages accepted by rebound automata with fc-dimensional input for k ≥ 3. Further we answer the question whether the classes of languages accepted by deterministic or nondeterministic rebound automata are closed under length-preserving homomorphisms negatively.