LAUSR

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 Ti:[0,1]→[0,1] be µ i -preserving ergodic transformations. We say that transformations T0,T1,…,Tk are uniformly jointly ergodic with respect to (λ;μ0,μ1,…,μk) if for any f0,f1,…,fk∈L∞ , limN−M→∞1N−M∑n=MN−1f0(T0 nx)⋅f1(T1 nx)⋯fk(Tk nx)=∏i=0k∫fidμi in L2(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, TG(x)=1x(mod1) , and, for β > 1, let T β denote the β-transformation defined by Tβx=βx(mod1) . Let T 0 be an ergodic interval exchange transformation. Let β1,…,βk be distinct real numbers with βi>1 and assume that logβi≠π26log2 for all i=1,2,…,k . Then for any f0,f1,f2,…,fk+1∈L∞(λ) , limN−M→∞1N−M∑n=MN−1f0(T0nx)⋅f1(Tβ1nx)⋯fk(Tβknx)⋅fk+1(TGnx)=∫f0dλ⋅∏i=1k∫fidμβi⋅∫fk+1dμGin L2(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.