Dynamics of piecewise isometries

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Statistical inference using weak chaos and infinite memory (2010, background): “It has been rigorously proven that the growth rate must have an exponent less or equal than K (Goetz, 2000), but we conjecture based on the above argument and some preliminary numerical simulations that it is actual equal to K for herding in the typical case (a.”
Herding Dynamic Weights for Partially Observed Random Field Models (2009, neutral): “In fact, the herding equations are similar to dynamical systems studied in the mathematical literature known as “piecewise isometries” [2].”
Piecewise Isometries — An Emerging Area of Dynamical Systems (2003, background): “There is some experimental and theoretical evidence [17] that the dynamics of piecewise isometries in Rn for n > 1 is closer in nature to the complex dynamics of rational maps [8], then to one-dimensional interval exchanges.”
Parametric Herding (2010, background): “This means that the number of distinct subsequences of length T grows polynomially inT (Goetz, 1996), in contrast to the exponential growth for stochastic and fully chaotic systems which have non-vanishing topological entropy.”
Frontiers of chaotic advection (2014, background): “Interval exchange transformations (Keane, 1975) and their generalizations known as piecewise isometries (Goetz, 2000) have no hyperbolic behavior and can FIG.”