SRB Measures Of Non-uniformly Expanding Mappings On S~1 And Skew-product Mappings On S~1×Ⅰ

Consider C¹ skew-product mapping f:S¹Ã—[0,1]→S¹Ã—[0,1], f(r, s)=(3r mod 1,f_r(s))-Let us denote by dr and ds the Lebesgue measure on S¹ and [0,1].Then the Lebesgue measure on S¹Ã—[0,1] is Leb=dr×ds. Notice thatμ₀ is the Lebesgue measure on S¹Ã—{0}, andμ₁ is the Lebesgue measure on S¹Ã—{1}. Naturally,μ₀ andμ₁ are the f-invariant ergodic measures of f.We assume f satisfies the following hypotheses:·Let r∈S¹, for any s G [0,1],0<|Df_r(s) |<3,f_r(0)=0,f_r(1)= 1. Here, Df_r(s) is the derivate of s and s∈[0,1].·Notice that 0 and 1/2 are the fixed pionts of S¹â†’S¹, r(?)3r mod 1. Then (0,0), (0,1), (1/2,0), (1/2,1) are the fixed pionts of f.·We assume that f₀ and f_1/2 have no fixed pionts in (0,1), and (f₀- id)|(0,1)<0; (f_1/2-id)|(0,1)>0;Df₀(0)<1;Df_1/2(1)<1.Thenμ₀ andμ₁ are the SRB measures of f. They satisfyLeb(B(μ₀)∪B(μ₁))=1,andHere, B(μ₀) is the basin ofμ₀, and B(μ₁) is the basin ofμ₁.Kan proved the conclusion with respect to C² function f(r, s)=(3r, s+cos(2Ï€r)(s/(32))(1-s)). Bonatti also proved the theorem in Chengxing math center, if the fountion is C². Though Bonatti's proof, the most important is that the distortion theorem is valid under the hypothese C². Moreover, we can find a local stable manifold.In this essay, for C¹ function family, one main novelty here is Pliss lemma. By Pliss lemma, we firstly get that forμ₀ andμ₁ everywhere x∈S¹Ã—[0,1], x has infinitely many hyperbolic times. Moreover, we insure the existence of local stable manifold and get the conclusion.