Chaos and Fractals around Black Holes

Fractal basin boundaries provide an important means of characterizing chaotic systems. We apply these ideas to general relativity, where other properties such as Lyapunov exponents are difficult to define in an observer independent manner. Here we discuss the difficulties in describing chaotic systems in general relativity and investigate the motion of particles in two- and three-black-hole spacetimes. We show that the dynamics is chaotic by exhibiting the basins of attraction of the black holes which have fractal boundaries. Overcoming problems of principle as well as numerical difficulties, we evaluate Lyapunov exponents numerically and find that some trajectories have a positive exponent.Comment: To appear in "Fractals" March issue (World Scientific), 20 figures available by request, also available from SLAC's gr-qc postscript archiv

Microscopic chaos and diffusion

We investigate the connections between microscopic chaos, defined on a dynamical level and arising from collisions between molecules, and diffusion, characterized by a mean square displacement proportional to the time. We use a number of models involving a single particle moving in two dimensions and colliding with fixed scatterers. We find that a number of microscopically nonchaotic models exhibit diffusion, and that the standard methods of chaotic time series analysis are ill suited to the problem of distinguishing between chaotic and nonchaotic microscopic dynamics. However, we show that periodic orbits play an important role in our models, in that their different properties in chaotic and nonchaotic systems can be used to distinguish such systems at the level of time series analysis, and in systems with absorbing boundaries. Our findings are relevant to experiments aimed at verifying the existence of chaoticity and related dynamical properties on a microscopic level in diffusive systems.Comment: 28 pages revtex, 14 figures incorporated with epsfig; see also chao-dyn/9904041; revised to clarify the definition of chaos and include discussion of a mixed model with both square and circular scatterer

Peeping at chaos: Nondestructive monitoring of chaotic systems by measuring long-time escape rates

One or more small holes provide non-destructive windows to observe corresponding closed systems, for example by measuring long time escape rates of particles as a function of hole sizes and positions. To leading order the escape rate of chaotic systems is proportional to the hole size and independent of position. Here we give exact formulas for the subsequent terms, as sums of correlation functions; these depend on hole size and position, hence yield information on the closed system dynamics. Conversely, the theory can be readily applied to experimental design, for example to control escape rates.Comment: Originally 4 pages and 2 eps figures incorporated into the text; v2 has more numerical results and discussion: now 6 pages, 4 figure