Dimensions of average conformal repeller.doc

http:/www.paper.edu.cnDimensions of average conformal repellerJungchao BanDepartment of mathematics National Hualien University of Education Hualien 97003, Taiwan jcbanmail.nhlue.edu.twYongluo CaoDepartment of mathematicsSuzhou UniversitySuzhou 215006, Jiangsu, P.R.China ylcaosuda.edu.cn, yongluocaoyahoo.comAbstract. In this paper, average conformal repeller is defined, which is generalization of conformal repeller. Using thermodynamic formalism for sub-additive potential defined in 5, Hausdorff dimension and box dimension of average conformal repellers are obtained. The map f is only needed C 1, without additional condition.Key words and phrases Hausdorff dimension, Non-conformal repellers, Topological pressure.1 Introduction.In the dimension theory of dynamical systems, and in particular in the study of the Hausdorff dimension of invariant sets of hyperbolic dynamics, the theory is only devel- oped to full satisfaction in the case of conformal dynamical systems (both invertible and non-invertible ). Roughly speaking, these are dynamical systems for which at each point the rate of contraction and expansion are the same in every direction. Bowen00 2000 Mathematics Subject classification: Primary 37D35; Secondary 37C45.133 was the first to express the Hausdorff dimension of an invariant set as a solution of an equation involving topological pressure. Ruelle 13 refined Bowens method andobtained the following result. Assume that f is a C 1+ conformal expanding map, isan isolated compact invariant set and f | is topologically mixing, then the Hausdorff dimension of , dimH is given by the unique solution of the equationP (f | , log kDxf k) = 0(1.1)where P (f | , ·) is the topological pressure functional. The smoothness C 1+ was re- cently relaxed to C 1 10.For non-conformal dynamical systems there exists only partial results. For example, the Hausdorff dimension of hyperbolic invariant sets was only computed in some specialcases. Hu 12 gave an estimate of dimension of non-conformal repeller for C 2 map.Falconer 7, 8 computed the Hausdorff dimension of a class of non-conformal repellers. Related ideas were applied by Simon and Solomyak 15 to compute the Hausdorffdimension of a class of non-conformal horseshoes in R3.For C 1 non-conformal repellers, in 17, the author uses singular values of the deriva-tive Dxf n for all n Z +, to define a new equation which involves the limit of a sequenceof topological pressure. Then he shows that the unique solution of the equation is anupper bounds of Hausdorff dimension of repeller. In 1, the same problem is con- sidered. The author bases on the non-additive thermodynamic formalism which wasintroduced in 2 and singular value of the derivative Dxf n for all n Z +, and givesan upper bounds of box dimension of repeller under the additional assumptions for which the map is C 1+ and -bunched. This automatically implies that for Hausdorffdimension. In 9, the author defines topological pressure of sub-additive potential un- der the condition k(Dxf )1k2kDxf k < 1, which means that f is 1bunched. Theyalso obtain an upper bounds of Hausdorff dimension of repeller. In 4, the first named author prove that the upper bounded of Hausdorff dimension for C 1 non-conformalrepeller obtained in 1, 9, 17 are same and it is the unique root of Bowen equation for sub-additive topological pressure.In this paper, we introduce the notion of average conformal repeller. Using thermo- dynamic formalism for sub-additive potential defined in 5, we prove that Hausdorff dimension and box dimension of average conformal repellers is the unique root of Bowenequation for subadditive topological pressure. The map f is only needed C 1, withoutadditional condition. Meanwhile, we introduce sup-additive potential topological pres- sure and prove that for special potentials, sub-additive and sup-additive topological pressures are same. In 2, 11, the authors introduce the concepts of quasi-conformal and asymptotically conformal repeller by using Markov construction and prove that its dimension is the unique root of the equation obtained by non-additive topological pressure. It is obvious that quasi-conformal and asymptotically conformal repeller are average conformal repellers, but reverse is not true. Therefore our result is a general- ization of the results in 2, 11.First we recall some basic definitions and notations.Let f : X X be a continuous map. A set E X is called (n, ) separated setiiwith respect to f if x, y E then dn(x, y) = max0in1 d(f x, f y) > . For x Xand r > 0, defineBn(x, r) = y X : f iy B(f ix, r), for all i = 0, · · · , n 1.If is a real continuous function on X and n Z +, letn1Sn(x) = X (f i(x).i=0We definePn(, , ) = supX exp Sn(x) : E is a (n, ) separated subset of X .xEThen the topological pressure of is given by1P (f, ) = lim lim suplog Pn(, ).0n nNext we give some properties of P (f, ·) : C (M, R) R .Proposition 1.1. Let f : M M be a continuous transformation of a compact metrisable space M . If 1 , 2 C (X, R), then the followings are true:(1) P (f, 0) = htop(f ).(2) |P (f, 1) P (f, 2)| k1 2 k.(3) 1 2 implies that P (f, 1) P (f, 2).Proof. See Walters book 16.Corollary 1. Let f : M M be a continuous transformation of a compact metrisable space M . If C (M, R) and < 0 then function P () = P (f, ) is continuous and strictly decreasing in .Proof. It easily follows from Proposition 1.1.The paper is organized as follows. In Section 2, we develop variational principal for sub-additive potential. In Section 3, we introduce the definition of average confor- mal repeller and give related results and the main theorem. In section 4, we develop sup-additive thermodynamics formalism and variational principal for sup-additive po- tential. In section 5, we give the proof of main result.2 A sub-additive thermodynamics formalismLet f : X X be a continuous map. A set E X is called (n, ) separated set withiirespect to f if x, y E then dn(x, y) = max0in1 d(f x, f y) > . A sub-additivevaluation on X is a sequence of functions n : M R such thatm+n(x) n(x) + m(f n(x),we denote it by F = n.In the following we will define the topological pressure of F = n with respect tof . We definePn(F , ) = sup XxEexp n(x) : E is a (n, ) separated subset of X .Then the topological pressure of F is given by1P (f, F ) = lim lim suplog Pn(F , ).0n nLet M(X ) be the space of all Borel probability measures endowed with the weak* topology. Let M(X, f ) denote the subspace of M(X ) consisting of all f -invariant measures. For µ M(X, f ), let hµ(f ) denote the entropy of f with respect to µ, and let F(µ) denote the following limit1 ZF(µ) = limndµ.n nThe existence of the above limit follows from a sub-additive argument. We call F(µ) the Lyapunov exponent of F with respect to µ since it describes the exponentially increasing speed of n with respect to µ.In 5, authors proved that the following variational principalTheorem 2.1. 5 Under the above general setting, we haveP (f, F ) = suphµ(T ) + F(µ) : µ M(X, f ).3 Average conformal repellerLet M be a C Riemann manifold, dim M = m. Let U be an open subset of M and let f : U M be a C 1 map. Suppose U is a compact invariant set, that is, f = and there is k > 1 such that for all x and v TxM ,kDxf vk kkvk,where k.k is the norm induced by an adapted Riemannian metric. Let M(f | ), E (f ) denote the all f invariant measures and the all ergodic invariant measure supported on respectively. By the Oseledec multiplicative ergodic theorem, for any µ E(f ), we can define Lyapunov exponents 1(µ) 2(µ) · · · n(µ), n = dimM .Definition 3.1. An invariant repeller is called average conformal if for any µ E(f ),1(µ) = 2(µ) = · · · = n(µ) > 0.It is obvious that a conformal repeller is an average conformal repeller, but reverse isnt true.Next we will give main theorem.Theorem 3.1. (Main Theorem) Let f be C 1 dynamical system and be an average conformal repeller, then the Hausdorff dimension of is zero t0 of t 7 P (tF ), whereF = log(m(Dxf n), x , n N.(3.2)where m(A) = kA1k1The proof will be given in section 5.Theorem 3.2. If be an average conformal repeller, then1uniformly on .limn n(log kDf n(x)k log m(df n(x) = 0Proof. LetFn(x) = log kDf n(x)k log m(df n(x), n N, x .It is obviously that the sequence Fn(x) is a non-negative subadditive function se- quence. That is sayFn+m(x) Fn(x) + Fm(f n(x), x .Suppose (3.2) is not true, then there exists 0 > 0, for any k N, there exits nk kand xnk such thatDefine measures1Fnk (xnk ) 0.nknk 1n1µnk =kXki=0f i (xn ).Compactness of P (f ) implies there exists a subsequence of µnk that converges to mea- sure µ. Without loss of generality, we suppose that µnk µ. It is well known that µ is f -invariant. Therefore µ M(f ).For a fixed m, we havelimkZ1Fm(xnk )dµnk =M mZ1Fm(xnk )dµ.M mIt implieslimnk 1X1Fm(f i(xnZ) =1mjFm(x)dµ.1k nkkmi=0 MFor a fixed m, let nk = ms + l, 0 l < m. The sub-additivity of Fn implies that forj = 0, · · · m 1,jFnk (xnk ) Fj (xnk ) + Fm(f(xnk ) + · · · + Fm(fm(s2)f(xnk )j+Fmj+l (fSumming j from 0 to m 1, we getm(s1)f(xnk )Fnk (xnk ) m1 s2X X1mFm(fim+jk(xnk )j=01 m1i=0+XFj (xnmkj=0) + Fmj+l (f (s1)m+j (xn)Let C1 = maxi=1,··· ,2m1 maxx Fi(x).(sm+l)1sm1Fnk (x) Xj=01nk 1X 1Fm(f mjj1 (x) mXj=(s1)mFm(fj (x) + 2C1Hence we havej=0Fm(f m(x) + 4C1.nk 1 Zlim1MFnk (x) lim1 X 1Fm(f i(x) =1Fm(x)dµ.k nkk nkm mi=0The arbitrariness of m N implies thatlim11 ZFnk (x) Fm(x)dµ, m N.Hencek nk m M1 Zlimm mFm(x)dµ 0 > 0.MThen ergodic decomposition theorem 16 implies that there exists µ E (f ) such that1 ZlimFm(x)dµ 0 > 0.m m MOn the other hand, from Oseledec theorem and Kingmans subadditive ergodic the-orem, we have lim 1 Rlog kDf n(x)kdµ= n(µ) and lim 1 Rlog m(f n(x)dµ =m m M1(µ). Thereforen(µ) 1 (µ) 0.m m MThis gives a contradiction to assumption of average conformal.4 Sup-additive variational principalIn this section, we first give the definition of sup-additive topological principal. Then we prove the variational principal for special sup-additive potential.Let f : X X be a continuous map. A set E X is called (n, ) separated set withiirespect to f if x, y E then dn(x, y) = max0in1 d(f x, f y) > . A sup-additivevaluation on X is a sequence of functions n : M R such thatm+n(x) n(x) + m(f n(x),we denote it by F = n.In the following we will define the topological pressure of F = n with respectto f . We defineP Xn (F , ) = supxEexp n(x) : E is a (n, ) separated subset of X .Then the topological pressure of F is given by1P (f, F ) = lim lim suplog Pn(F , ).0n nFor every µ M(X, f ), let F(µ) denote the following limit1 ZF(µ) = limn nndµ.The existence of the above limit follows from a sup-additive argument. We call F(µ)the Lyapunov exponent of F with respect to µ since it describes the exponentiallyincreasing speed of n with respect to µ.Theorem 4.1. Let f be C 1 dynamical system and be an average conformal repeller, and F = n(x) = t log kDf n(x)k for t 0 be a sup-additive function sequence.Then we haveP (f, F ) = suphµ(T ) + F(µ) : µ M(X, f ).Proof. First we prove that for any m NP (f, F ) P (f,m ). mFor a fixed m, let n = ms + l, 0 l < m. From the sup-additivity of n, we have1n(x) mm1 s2X Xm(fim+j1 (x) +mm1Xj (x) + mj+l (f(s1)m+j(x).j=0i=0j=0Let C1 = mini=1,··· ,2m1 minxX i(x). Then it has(sm+l)1 11sm1n(x) X m(f j (x) mj=0n1Xmj=(s1)mm(f j (x) + 2C1m m X 1 (fj=0j (x) + 4C1.Hence we haven1 1mexp(n(x) exp(X m(f j (x) + 4C1 ).j=0ThusPn (F , )=supX exp n(x) : E is a (n, ) separated subset of X It impliesxE1 Pn( m m, ) × exp(4C1).1P (f, F ) P (f, m m) .From the arbitrary of m Z +, we have1+P (f, F ) P (f, m m), for all m Z .By the variational principal in 16, for every µ M(f ), we have1ZmP (f, F ) P (f,m) hµ(f ) +mMHence we have for every µ M(f )1n(x)dµ, m N.ThereforemP (f, F ) hµ(f ) + limZZ1n(x)dµ.M m1mP (f, F ) suphµ(f ) + limn(x)dµ, µ M(f )mMLet n(x) = t log m(Df n(x) for t 0. Then it is sub-additive. By the theorem in 5, we havemP (f, n) = suphµ(f ) + limZ1n(x)dµ, µ M(f )mMBy the definitions, t log m(Df n(x) t log kDf n(x)k for t 0 implies thatP (f, F ) P (f, n).Theorem 3.2 implies that for any µ M(f ), it hasThereforelimmZ1n(x)dµ = limM m mZ1n(x)dµ.M mmP (f, F ) = suphµ(f ) + limThis completes the proof of theorem.Z1n(x)dµ, µ M(f ).mM5 The proof of main theoremIn this section, we will give the proof of main theorem. First we state some known results.In 1, Barreira prove the following theorem.Theorem 5.1. If f is a C 1 expanding map and is a repeller, thens1 dimH dimB dimB t1