Follow the fugitive: an application of the method of images to open systems

Abstract

Borrowing and extending the method of images we introduce a theoretical framework that greatly simplifies analytical and numerical investigations of the escape rate in open systems. As an example, we explicitly derive the exact size-and position-dependent escape rate in a Markov case for holes of finite size. Moreover, a general relation between the transfer operators of the closed and corresponding open systems, together with the generating function of the probability of return to the hole is derived. This relation is then used to compute the small hole asymptotic behavior, in terms of readily calculable quantities. As an example we derive logarithmic corrections in the second order term. Being valid for Markov systems, our framework can find application in many areas of the physical sciences such as information theory, network theory, quantum Weyl law and via Ulam's method can be used as an approximation method in general dynamical systems.

Escape rates for special flows and their higher order asymptotics

“…Note that this method requires m ≤ n. If m > n, one could move to a larger refined alphabet, but this would entail an exponential growth of the size of the matrix representation. A better way of representing L op for small holes was devised by Cristadoro, Knight and Degli Esposti [CKDE13]. The remainder of this subsection summarises the relevant results when adapted to the arithmetic special flow situation; a more detailed account can be found in [Dre15, Kapitel 7.1.1].…”

“…Continuing in the setting of the previous subsection, one considers a sequence of shrinking holes. The corresponding ideas in [CKDE13] are adapted to accommodate the shape of the underlying special flow. Fix a periodic point x = (x 1 , x 2 , .…”

“…The idea for obtaining an approximation of the zero describing the escape rate that was laid out at the end of the previous subsection will be carried out. In contrast to [CKDE13], the argument will be carried out rigorously and ultimately yield a little-o statement regarding the quality of the approximation.…”

“…Finally, the most restrictive setting is used in Section 7 where the base transformation is required to be a Markov shift on a finite alphabet and the ceiling function is required to be constant on cylinder sets of a certain length. In this section, the results of Cristadoro, Knight and Degli Esposti [CKDE13] are expanded upon in order to prove higher order asymptotics for the escape rate of shrinking holes (Theorems 7.8 and 7.9). This also allows one to recover information about the orbit lengths of periodic points purely by considering escape rates and the measure of shrinking holes.…”

Some new surprises in chaos

“…5,7,8,13 Indeed, if a set A contains a periodic orbit of period n, then l(A \ T n A) is the probability to return to A rather than to hit A for the first time at the moment n. Clearly, lðAÞ ¼ lðH n ðAÞÞ þ lðR n ðAÞÞ, where R n ðAÞ ¼ fx : x 2 A; T n x 2 Ag. Therefore, the faster the orbits return to A, the less the first hitting probability will be.…”