Let $$f:\mathbb {M}\rightarrow \mathbb {M}$$
f
:
M
→
M
be a continuous map on a compact metric space $$\mathbb {M}$$
M
equipped with a fixed metric d, and let $$\tau $$
τ
be the topology on $$\mathbb {M}$$
M
induced by d. We denote by $$\mathbb {M}(\tau )$$
M
(
τ
)
the set consisting of all metrics on $$\mathbb {M}$$
M
that are equivalent to d. Let $$ \text {mdim}_{\text {M}}(\mathbb {M},d, f)$$
mdim
M
(
M
,
d
,
f
)
and $$ \text {mdim}_{\text {H}} (\mathbb {M},d, f)$$
mdim
H
(
M
,
d
,
f
)
be, respectively, the metric mean dimension and mean Hausdorff dimension of f. First, we will establish some fundamental properties of the mean Hausdorff dimension. Furthermore, it is important to note that $$ \text {mdim}_{\text {M}}(\mathbb {M},d, f)$$
mdim
M
(
M
,
d
,
f
)
and $$ \text {mdim}_{\text {H}} (\mathbb {M},d, f)$$
mdim
H
(
M
,
d
,
f
)
depend on the metric d chosen for $$\mathbb {M}$$
M
. In this work, we will prove that, for a fixed dynamical system $$f:\mathbb {M}\rightarrow \mathbb {M}$$
f
:
M
→
M
, the functions $$\text {mdim}_{\text {M}} (\mathbb {M}, f):\mathbb {M}(\tau )\rightarrow \mathbb {R}\cup \{\infty \}$$
mdim
M
(
M
,
f
)
:
M
(
τ
)
→
R
∪
{
∞
}
and $$ \text {mdim}_{\text {H}}(\mathbb {M}, f): \mathbb {M}(\tau )\rightarrow \mathbb {R}\cup \{\infty \}$$
mdim
H
(
M
,
f
)
:
M
(
τ
)
→
R
∪
{
∞
}
are not continuous, where $$ \text {mdim}_{\text {M}}(\mathbb {M}, f) (\rho )= \text {mdim}_{\text {M}} (\mathbb {M},\rho , f)$$
mdim
M
(
M
,
f
)
(
ρ
)
=
mdim
M
(
M
,
ρ
,
f
)
and $$ \text {mdim}_{\text {H}}(\mathbb {M}, f) (\rho )= \text {mdim}_{\text {H}} (\mathbb {M},\rho , f)$$
mdim
H
(
M
,
f
)
(
ρ
)
=
mdim
H
(
M
,
ρ
,
f
)
for any $$\rho \in \mathbb {M}(\tau )$$
ρ
∈
M
(
τ
)
. Furthermore, we will present examples of certain classes of metrics for which the metric mean dimension is a continuous function.
