  It is shown that the description of anomalous scaling in turbulent systems
requires the simultaneous use of two normalization scales. This phenomenon
stems from the existence of two independent (infinite) sets of anomalous
scaling exponents that appear in leading order, one set due to infrared
anomalies, and the other due to ultraviolet anomalies. To expose this clearly
we introduce here a set of local fields whose correlation functions depend
simultaneously on the the two sets of exponents. Thus the Kolmogorov picture of
"inertial range" scaling is shown to fail because of anomalies that are
sensitive to the {\em two ends} of this range.
