Let
S
=
{
S
1
,
…
,
S
K
}
{\boldsymbol {S}}=\{S_1,\ldots ,S_K\}
be a finite set of complex
d
×
d
d\times d
matrices and
Σ
K
+
\varSigma _{\!K}^+
be the compact space of all one-sided infinite sequences
i
⋅
:
N
→
{
1
,
…
,
K
}
i_{\boldsymbol {\cdot }}\colon \mathbb {N}\rightarrow \{1,\dotsc ,K\}
. An ergodic probability
μ
∗
\mu _*
of the Markov shift
θ
:
Σ
K
+
→
Σ
K
+
;
i
⋅
↦
i
⋅
+
1
\theta \colon \varSigma _{\!K}^+\rightarrow \varSigma _{\!K}^+;\ i_{\boldsymbol {\cdot }}\mapsto i_{\boldsymbol {\cdot }+1}
, is called “extremal” for
S
{\boldsymbol {S}}
if
ρ
(
S
)
=
lim
n
→
∞
‖
S
i
1
⋯
S
i
n
‖
n
{\rho }({\boldsymbol {S}})=\lim _{n\to \infty }\sqrt [n]{\left \|S_{i_1}\cdots S_{i_n}\right \|}
holds for
μ
∗
\mu _*
-a.e.
i
⋅
∈
Σ
K
+
i_{\boldsymbol {\cdot }}\in \varSigma _{\!K}^+
, where
ρ
(
S
)
\rho ({\boldsymbol {S}})
denotes the generalized/joint spectral radius of
S
{\boldsymbol {S}}
. Using the extremal norm and the Kingman subadditive ergodic theorem, it is shown that
S
{\boldsymbol {S}}
has the spectral finiteness property (i.e.
ρ
(
S
)
=
ρ
(
S
i
1
⋯
S
i
n
)
n
\rho ({\boldsymbol {S}})=\sqrt [n]{\rho (S_{i_1}\cdots S_{i_n})}
for some finite-length word
(
i
1
,
…
,
i
n
)
(i_1,\ldots ,i_n)
) if and only if for some extremal measure
μ
∗
\mu _*
of
S
{\boldsymbol {S}}
, it has at least one periodic density point
i
⋅
∈
Σ
K
+
i_{\boldsymbol {\cdot }}\in \varSigma _{\!K}^+
.