ON INFINITE MEASURE PRESERVING MEASURABLE DYNAMICAL SYSTEMS

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The following three are the main results of this project:An unite measure preserving transformation can be reduced to a finite measure preserving one as an induced transformation with a set of finite measure. In this case, the ceiling function represents the return time to this set and it is non-integrable. In this sense, we studied the stochastic processes with infinite expectations. In particular, we studied continued fraction mixing stochastic processes with barely infinite expectations. We have proved the strong law of large numbers after light trimming under some conditions on the distribution of random variables.We also studied Maharum extension of ergodic nonsingular transformations. We discussed locally finite invariant measures for Maharam extensions associated to irrational rotations and subshifts. In the case of finite Markov subshifts, we characterized such measures by conformal measures for subshifts. We extend such results to countable Markov subshifts.As an application of infinite ergodic theory, we studied the theory of arithmetic progressions in particular, concerning to Erods conjecture. In this point of view, we considered the multiple recurrence property of infinite measure preserving transformations. We gave an lower estimate of the multiplicity by Kakutani type. We also studied metric number theory as an application of ergodic theory. Some of main results are the following (1) We proved a property of the arithmetic distribution of convergents arising from the Jacobi-Perron algorithm. (2) We constructed Farey maps associated to Rosen's continued fractions. here the Farey maps are 1-dimensional maps which induces mediant convergents of continued fractions.