Riddling and invariance for discontinuous maps preserving Lebesgue measure

In this paper we use the mixture of topological and measure-theoretic dynamical approaches to consider riddling of invariant sets for some discontinuous maps of compact regions of the plane that preserve two-dimensional Lebesgue measure. We consider maps that are piecewise continuous and with invertible except on a closed zero measure set. We show that riddling is an invariant property that can be used to characterize invariant sets, and prove results that give a non-trivial decomposion of what we call partially riddled invariant sets into smaller invariant sets. For a particular example, a piecewise isometry that arises in signal processing (the overflow oscillation map), we present evidence that the closure of the set of trajectories that accumulate on the discontinuity is fully riddled. This supports a conjecture that there are typically an infinite number of periodic orbits for this system.

Multiple maps and activity-dependent representational plasticity in the anterior Wulst of the adult barn owl (Tyto alba)

In the present study we addressed the issue of somatosensory representation and plasticity in a nonmammalian species, the barn owl. Multiunit mapping techniques were used to examine the representation of the specialized receptor surface of the claw in the anterior Wulst. We found dual somatotopic mirror image representations of the skin surface of the contralateral claw. In addition, we examined both representations 2 weeks after denervation of the distal skin surface of a single digit. In both representations, the denervated digital representation became responsive to stimulation of the adjacent, mutually functional, digit. The mutability and multiple representations indicates that the Wulst provides the owl with sensory processing capabilities analogous to those in mammals.

Pseudo-randomness of round-off errors in discretized linear maps on the plane

We analyze the sequences of round-off errors of the orbits of a discretized planar rotation, from a probabilistic angle. It was shown [Bosio & Vivaldi, 2000] that for a dense set of parameters, the discretized map can be embedded into an expanding p-adic dynamical system, which serves as a source of deterministic randomness. For each parameter value, these systems can generate infinitely many distinct pseudo-random sequences over a finite alphabet, whose average period is conjectured to grow exponentially with the bit-length of the initial condition (the seed). We study some properties of these symbolic sequences, deriving a central limit theorem for the deviations between round-off and exact orbits, and obtain bounds concerning repetitions of words. We also explore some asymptotic problems computationally, verifying, among other things, that the occurrence of words of a given length is consistent with that of an abstract Bernoulli sequence.