Thermodynamics of Lorenz-type maps.

The thermodynamic properties of maps derived from strongly dissipative Lorenz-type flows are studied by means of generalized Frobenius-Perron operators. Three different types of phase transitions are found: (I) second-order transitions occurring at zero temperatures caused by short-range interactions in an associated spin chain characterizing hyperbolic maps, (II) first-order transitions caused by long-range interactions at finite temperatures induced by anomalous but exponential scaling in nonhyperbolic cases, and (III) transitions of extreme orders induced by nonexponential scaling in certain nonhyperbolic maps. Type-III transitions might also appear in the spectra of generalized entropies at nonzero temperatures. A novel feature of Lorenz-type maps is the existence of type-II-type-III and type-III-type-III double transitions.

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