There are several versions of the Kaplan-Yorke conjecture, with many of the higher dimensional ones remaining unsettled. The original Kaplan-Yorke conjecture (Kaplan
and Yorke 1979) proposed that, for a two-dimensional mapping, the capacity
dimension 
equals the Kaplan-Yorke dimension 
,
 |
where 
and 
are the Lyapunov characteristic exponents.
This was subsequently proven to be true in 1982. A later conjecture held that the
Kaplan-Yorke dimension is generically equal
to a probabilistic dimension which appears to be identical to the information
dimension (Frederickson et al. 1983). This conjecture is partially verified
by Ledrappier (1981). For invertible two-dimensional maps, 
, where 
is the correlation exponent,
is the information dimension, and 
is the capacity dimension
(Young 1984).
