Abstract
Abstract
For
i
=
0
,
1
,
2
,
…
,
k
, let µ
i
be a Borel probability measure on
[
0
,
1
]
which is equivalent to the Lebesgue measure λ and let
T
i
:
[
0
,
1
]
→
[
0
,
1
]
be µ
i
-preserving ergodic transformations. We say that transformations
T
0
,
T
1
,
…
,
T
k
are uniformly jointly ergodic with respect to
(
λ
;
μ
0
,
μ
1
,
…
,
μ
k
)
if for any
f
0
,
f
1
,
…
,
f
k
∈
L
∞
,
lim
N
−
M
→
∞
1
N
−
M
∑
n
=
M
N
−
1
f
0
(
T
0
n
x
)
⋅
f
1
(
T
1
n
x
)
⋯
f
k
(
T
k
n
x
)
=
∏
i
=
0
k
∫
f
i
d
μ
i
in
L
2
(
λ
)
.
We establish convenient criteria for uniform joint ergodicity and obtain numerous applications, most of which deal with interval maps. Here is a description of one such application. Let T
G
denote the Gauss map,
T
G
(
x
)
=
1
x
(
m
o
d
1
)
, and, for β > 1, let T
β
denote the β-transformation defined by
T
β
x
=
β
x
(
m
o
d
1
)
. Let T
0 be an ergodic interval exchange transformation. Let
β
1
,
…
,
β
k
be distinct real numbers with
β
i
>
1
and assume that
log
β
i
≠
π
2
6
log
2
for all
i
=
1
,
2
,
…
,
k
. Then for any
f
0
,
f
1
,
f
2
,
…
,
f
k
+
1
∈
L
∞
(
λ
)
,
lim
N
−
M
→
∞
1
N
−
M
∑
n
=
M
N
−
1
f
0
(
T
0
n
x
)
⋅
f
1
(
T
β
1
n
x
)
⋯
f
k
(
T
β
k
n
x
)
⋅
f
k
+
1
(
T
G
n
x
)
=
∫
f
0
d
λ
⋅
∏
i
=
1
k
∫
f
i
d
μ
β
i
⋅
∫
f
k
+
1
d
μ
G
in
L
2
(
λ
)
.
We also study the phenomenon of joint mixing. Among other things we establish joint mixing for skew tent maps and for restrictions of finite Blaschke products to the unit circle.
Funder
National Research Foundation of Korea
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics