Let D be a bounded domain in the complex plane, and let
ζ
\zeta
belong to the topological boundary
∂
D
\partial D
of D. We prove two theorems concerning the cluster set
Cl
(
f
,
ζ
)
{\text {Cl}}(f,\zeta )
of a bounded analytic function f on D. The first theorem asserts that values in
Cl
(
f
,
ζ
)
∖
f
(
Ш
ζ
)
{\text {Cl}}(f,\zeta )\backslash f(\Sha _\zeta )
are assumed infinitely often in every neighborhood of
ζ
\zeta
, with the exception of those lying in a set of zero analytic capacity. The second asserts that all values in
Cl
(
f
,
ζ
)
∖
f
(
M
ζ
∩
supp
λ
)
{\text {Cl}}(f,\zeta )\backslash f({\mathfrak {M}_\zeta } \cap {\text {supp}}\;\lambda )
are assumed infinitely often in every neighborhood of
ζ
\zeta
, with the exception of those lying in a set of zero logarithmic capacity. Here
M
ζ
{\mathfrak {M}_\zeta }
is the fiber of the maximal ideal space
M
(
D
)
\mathfrak {M}(D)
of
H
∞
(
D
)
{H^\infty }(D)
lying over
ζ
\zeta
,
Ш
ζ
\Sha _\zeta
is the Shilov boundary of the fiber algebra, and
λ
\lambda
is the harmonic measure on
M
(
D
)
\mathfrak {M}(D)
.