On Variations of the SRB Entropy of the Expanding Map on the Circle

Many people are familiar with the geometrical shape called the circle. Based on this figure, the circle space S connects the endpoints of the interval [0, 1] together so that 0 ≡ 1 (mod 1). On this space, the expanding map f : S → S stretches an initial distribution of points along the circle and then rewraps the lengthened distribution tightly about the circle; overlapping regions are compressed together to yield the new distribution along the circle. By iteratively performing the expanding map, we get a discrete dynamical system whose orbits are chaotic. The entropy is an observation of the complexity of this chaos with respect to a given measure. We study variations of the entropy of expanding maps that are small perturbations of the uniformly expanding map on S with respect to the physical measure, also known as the Sinai Ruelle Bowen (SRB) invariant measure. Due to the complexity of these computations, we discuss methods for computing this entropy numerically and approximate the entropy for oneand two-parameter variants of the expanding map.