  Along with widely used numerical methods for estimating and computing the
Lyapunov dimension there is an effective analytical approach, proposed by G.A.
Leonov in 1991. The Leonov method is based on the direct Lyapunov method with
special Lyapunov-like functions. The advantage of this method is that it allows
one to estimate the Lyapunov dimension of invariant set without local- ization
of the set in the phase space and in many cases get effectively exact Lyapunov
dimension formula. In this survey the invariance of Lyapunov dimension with
respect to diffeomorphisms and its connection with the Leonov method are
discussed. An analog of Leonov method for discrete time dynamical systems is
suggested. In a simple but rigorous way, here it is presented the connection
between the Leonov method and the key related works in the area: by Kaplan and
Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterle (upper
bounds of Hausdorff dimension via the Lyapunov dimension of maps, 1980),
Constantin, Eden, Foias, and Temam (upper bounds of Hausdorff dimension via the
Lyapunov exponents and dimension of dynamical systems, 1985-90), and the
numerical calculation of the Lyapunov exponents and dimension.

Chaotic Dynamics