Abstract
In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire’s sense) invariant measure has, for each
q
>
0
{q>0}
, zero lower q-generalized fractal dimension.
This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence.
Of special interest is the full-shift system
(
X
,
T
)
{(X,T)}
(where
X
=
M
ℤ
{X=M^{\mathbb{Z}}}
is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each
q
>
1
{q>1}
, infinite upper q-correlation dimension.
Under the same conditions, we show that a typical invariant measure has, for each
s
∈
(
0
,
1
)
{s\in(0,1)}
and each
q
>
1
{q>1}
, zero lower s-generalized and infinite upper q-generalized dimensions.
A Note on the Relation Between the Metric Entropy and the Generalized Fractal Dimensions of Invariant Measures
