A class of linear transformations

An Ergodic Theorem for Multidimensional Superadditive Processes

“…Moreover, for the results of our work, there is no loss of generality in assuming that U' 0 = /, the identity operator, where U' 0 belongs to the restriction of ^to # [2,3]. Hence one obtains the conclusions of Theorem 3.5 and Theorem 3.6 on fé 7 . It is obvious that in general there is no convergence on Sd.…”

“…If {~F( M ,v))( w ,v)ec is ^-superadditive, then {^( MiV) } (w , v)ec is called ^-subadditive; and' if both {-i 7 (M , v) } (M , v)ec and {^( M , v) } (M , v)GC are ^-superadditive, then {L V )L v)e( ; is called ^-additive [2,3].…”

“…Any strongly ^-superadditive process F = {^W V )L v)É(: which satisfies (1)(2)(3)(4)(5)(6)(7)(8) *io,v> = F («,0) = °> «>0, v>0…”

An ergodic theorem for non-invariant measures

“…The converse is not so straightforward and needs the full extent of the hypothesis, that is, that (II) holds for any f . Chacon described in [1] an example of a positive contraction U on some L 1 (X, µ) and a positive bounded and integrable function f such that…”

Ergodicity of the Cartesian product

“…Example 1. In [2] or [4, Example 6.8] a general method is given for constructing ergodic transformations that do not accept a finite invariant measure and in [ll] or [4, Example 6.8] a conservative and ergodic measure preserving transformation on a probability space is constructed having no eigenvalues except 1. It is possible to combine these two techniques and obtain an ergodic transformation that does not accept a finite invariant measure and also that has no eigenvalues except 1.…”