On ergodic and mixing properties of the triangle map

Abstract

In this paper we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters defining the map. The dynamics is studied numerically by means of two different symbolic encoding schemes, both relying on the fact that it maps polygons to polygons: in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons, in the second we consider the standard binary partition of the torus induced by the discontinuity set. These encoding schemes are studied in detail and shown to be compatible, although not equivalent. The ergodic properties of the triangle map are then investigated in terms of the Markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the Koopman operator in L2(T2). Finally, a stochastic version of the triangle map is introduced and studied. A simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map.Comment: 45 pages, 30 figure

Complexity and nonseparability of classical Liouvillian dynamics

“…1 we show time evolving phase space densities ρ −t for all four maps at t = 3, 5, 7, all starting from the same simple initial density ρ 0 (x, y) = (2 + cos x + cos y)(2π √ 5). Note that the three maps PC, SM and TM exhibit dynamical mixing behavior, although for TM the mixing mechanism is qualitatively different [10]. Only the orbits of the first two maps (PC and SM) have positive Kolmogorov complexity, with estimated Lyapunov exponents (being equal to Kolmogorov-Sinai entropies) λ PC max = 0.9496, λ SM max = 0.8206, and only PC exhibits exponential decay of correlations, d 2 zρ 0 (z)ρ t (z) − 1 ∼ exp(−ξt), with ξ PC = 1.17, while for SM and TM correlations decay as power laws.…”

Nonlinear rotations on a lattice

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Introduction to Nonextensive Statistical Mechanics

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“…We shall dedicate the present subsection to weakly chaotic twodimensional systems, namely the Casati-Prosen map (or triangle map) [8][9][10] and the Moore map [134][135][136], the former as focused on in [358], the latter as focused on in [138].…”