Statistical properties of equilibrium states for complex systems

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(1999)One of purposes of the project in 1999 is to clarify sufficient conditions for the existence of conformal measures for countable to one piecewise invertible Markov systems. About this problem, Prof.M.Denker gave valuable advices during his stay in Sapporo so that I could establish a new method for the construction of conformal measures which is based on the existence of a derived map T^* (Schweiger's jump transformation) which is uniformly expanding and guarantees a weak Holder-type property of the potential φ^* associated to φ. The result is contained in (3) which is a joint paper with M.Denker. Another purpose of the project is to establish bounds on decay of correlation functions for noninvertible maps with indifferent periodic points. I could obtained polynomial bounds by applying Liverani's method based on random parturbations of Perron-Frobenius operators and by estimating order of divergence of invariant densities near indifferent periodic points which was established in (1). The result is containaed in (4) which is a joint paper with M.Pollicott.(2000)In the second year project, I studied meromorphic properties of dynamical zeta functions for noninvertible maps with indifferent periodic points. I could clarify the meromorpic dmeain of the zeta functions by observing a good relation between the topological pressure for φ and the topological pressure associated to φ^* with respect to the jump transformation T^*. The result is contained in (5) which is a joint paper with M.Pollicott. Furthermore, I could improve the results on the rates of decay of correlations in (4) by clarifying the speed of uniform convergence of iterated Perron-Frobenius operators on compact sets excluding indifferent periodic points. The result is contained in (6).