Joint ergodicity of piecewise monotone interval maps

Abstract

For , let µ i be a Borel probability measure on which is equivalent to the Lebesgue measure λ and let be µ i -preserving ergodic transformations. We say that transformations are uniformly jointly ergodic with respect to if for any , 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, , and, for β > 1, let T β denote the β-transformation defined by . Let T 0 be an ergodic interval exchange transformation. Let be distinct real numbers with and assume that for all . Then for any , 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.