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DCDS

In this paper we consider an ergodic measure with bounded distortion on a symbolic space generated by an infinite alphabet, and showed that for each $r\in (0, +\infty)$ there exists a unique $k_r \in (0, +\infty)$ such that both the $k_r$-dimensional lower and upper quantization coefficients for its image measure $m$ with the support lying on the limit set generated by an infinite conformal iterated function system satisfying the strong open set condition are finite and positive. In addition, it shows that $k_r$ can be expressed by a simple formula involving the temperature function of the system. The result extends and generalizes a similar result of
Roychowdhury established for a finite conformal iterated function system [Bull. Polish Acad. Sci. Math. 57 (2009)].

JMD

One says that two ergodic systems $(X,\mathcal F,\mu)$ and $(Y,\mathcal
G,\nu)$ preserving a probability measure are evenly Kakutani equivalent if
there exists an orbit equivalence $\phi: X\to Y$ such that, restricted
to some subset $A\subseteq X$ of positive measure, $\phi$ becomes a
conjugacy between the two induced maps $T_A$ and $S_{\phi(A)}$. It follows
from the general theory of loosely Bernoulli systems developed in [8] that all adding machines are evenly Kakutani equivalent, as they
are rank-1 systems. Recent work has shown that, in systems that are
both topological and measure-preserving, it is natural to seek to
strengthen purely measurable results to be "nearly continuous''. In the
case of even Kakutani equivalence, what one asks is that the map $\phi$ and
its inverse should be continuous on $G_\delta$ subsets of full measure and
that the set $A$ should be within measure zero of being open and of being
closed. What we will show here is that any two adding machines are indeed
equivalent in this nearly continuous sense.

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