Rufus Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S. 50 (1979), 11--26.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the unique root of the equation P (sOE) 0: For the Moran constructions, s is the Hausdorff dimension of the limit set as well as its box dimension. We show this is true for a Moran construction which is modeled by an arbitrary symbolic process. The equation P (sOE) 0 was discovered by Bowen [B] and seems to be universal. We will show that all known equations previously used to compute the Hausdorff dimension can be deduced from this equation. For the general symbolic geometric constructions the measure m is an equilibrium measure and admits the non uniform mass distribution principle. ....

R. Bowen and C. Series, Hausdorff Dimension of Quasi-circles, Publ. Math. IHES 50 (1979), 11--25.

....Subject Classification. 58F20, 58F11, 11J70, 11J06. Typeset by A M S T E X 1 2 OLIVER JENKINSON AND MARK POLLICOTT A Schottky group Gamma = fl 1 ; fl n is a free group contained in PSL(2; C ) for which, in particular, the sets D(fl i ) fz 2 C : jfl 0 i (z)j 1g are disjoint (cf. [Bowen] and [P S] The limit set ae C is the set of accumulation points of the sequence fl i 1 : fl i k (0) say. The value d = dimH has various important interpretations relating to the corresponding surface of negative curvature. For example it equals the topological entropy of the geodesic flow ....

....The particular choice f(x) Gammas log jT 0 (x)j (restricted to E 2 ) gives the following result. Lemma 1. There is a unique solution 0 s 1 to P ( Gammas log jT 0 j) 0, which occurs precisely at s = dimH (E 2 ) Falconer] Bed] Formulae of this general type have a long history. Bowen [Bowen] showed that the dimension of quasi circles could be studied using thermodynamic ideas. Subsequently, Ruelle [Ru2] applied the approach to hyperbolic rational maps and McCluskey Manning [Mc M] studied Smale horseshoes. Our example is covered by the standard theory of cookie cutters (see for ....

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes ' Etudes Sci. Publ. Math. 50 (1979), 11--25.

....F on a Riemannian manifold M . If F is conformal with respect to some Riemannian metric then the Hausdorff dimension HD(J) of J can be obtained as the zero t 0 of t 7 F ( Gammat log kD x Fk) where F denotes the pressure functional. This statement is known as the Bowen Ruelle formula (cf. Bowen [Bow79], Ruelle [Rue82, Rue83] Moreover, one has that the equilibrium state of x 7 Gammat 0 log kD x Fk is equivalent to the t 0 Hausdorff measure. In this paper we consider a random perturbation of this situation which is modelled using the notion of a (bundle) random dynamical system (abbreviated ....

Rufus Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S. 50 (1979), 11--26.

....expanding map f : S k j=1 OE j (X) X defined by f(x) OE Gamma1 j (x) for x 2 OE j (X) Suppose that all the maps OE j are in C 1 (X) for some 2 (0; 1] and are hyperbolic, that is, 0 jOE 0 j (x)j 1 on X. Then the Hausdorff dimension dim H (J Phi ) is given by Bowen s formula [Bo2, R]: dim H (J Phi ) s( Phi) where P Phi (s( Phi) 0: Here P Phi (t) is the pressure function, which can be defined by P Phi (t) lim n 1 1 n log X 2I n k OE 0 k t where I = f1; kg, OE = OE 1 ffi Delta Delta Delta ffi OE n , and k Delta k is the supremum norm ....

R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S. 50 (1979), 11--26.

.... dimension is strictly between 0 and 2, that the Hausdorff measure at this dimension d is positive and finite, and moreover that = H d j J is geometric in the sense that 9c 1 ; c 2 0 with c 1 d (B (z) c 2 d for all sufficiently small (work of Bowen, Ruelle, Sullivan: [Bo2], Rue1] Su2] see also [PU] for general background on Julia sets see also [Be] CG] and [Mi] In this paper we will study a linear, continuous time dynamics which is constructed directly from the geometry of the set J . We imagine zooming toward some chosen point z 2 J . Now for a fractal ....

....density now follows as a corollary (see Proposition 4.2) The following theorem, describing the ergodic theory of the projected flow, follows from the fundamental work of Bowen, Ruelle and Sinai. Some of the main points in the developmant of the BRS theory relevant here are: Lemma 10 of [Bo2] (for Bowen s formula for Hausdorff dimension) x8 of [Si] and Proposition 3.1 of [BR] for the relationship between measures of maximal entropy for flows and Gibbs states on a cross section) For completeness we include the proof. In summary, two separate results from the BRS theory (the ....

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R. Bowen, Hausdorff dimension of quasi-circles, IHES Publ. Math. 50 (1980), 11-25.

.... ffi oe k ; 5) where the supremum is taken over the sequences (j 1 j 2 Delta Delta Delta ) 2 Q such that (j 1 Delta Delta Delta j n ) i 1 Delta Delta Delta i n ) As this example illustrates, we can use the thermodynamic formalism in complicated problems of dimension theory. In [Bo], Bowen introduced equation (4) It has a rather universal character: most equations used to compute or estimate dimensions are particular cases of it. For example, if Q = Sigma p then equation (4) is equivalent to equation (2) We can also consider geometric constructions modeled by ....

R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes ' Etudes Sci. Publ. Math. 50 (1979), 259--273.

.... X distance of the vertices having graph distance m to the origin O (these vertices correspond to the words of length m in F ) These inequalities give a computable way to find lower bounds for (F ) In particular we have the inequality (F ) sup 2E;h( 0 h( ff( 0:2) The works of Bowen [Bow] and Ruelle [Rue] suggest that in the case F is Fuchsian, or more generally when F is geometrically finite free group of hyperbolic isometries, then equality holds in (0.2) The maximal measure is the Bowen Margulis Ruelle measure which should be closely related to the Patterson Sullivan ....

....; m = 1; In particular (OE) sup 2E;h( 0 h( ff( 5:15) x6. Remarks Assume that F is a Fuchsian group that satisfies the conditions of Theorem 5.7. Then (F ) ffi(F ) and we conjecture that equality holds in (0. 2) The motivation for this conjecture are the results of Bowen [Bow] and Ruelle [Rue] For the simplicity of the exposition we consider Ruelle s results. Let f : CP CP be a rational function on the Riemann sphere of degree d 1. Denote by J(f) the Julia set of f . Assume that J(f) is a repeller. Set q(z) Gamma log jf 0 (z)j; z 2 J(f) Consider the ....

R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S., 50 (1979), 11-26.

....distances between o and other points of the orbit. We show dimH (F ) ffi(OE) ffi(OE) 0:7) Hence we can use lower bounds (0. 5) to get lower bounds on dimH (F ) Moreover, we show that these lower bounds are arbitrary close to dimH (F ) Our results are complementary to the results of Bowen [Bow2], who showed how to apply the thermodynamics formalism to the action of F on CP to find dimH (F ) In the last section we show how to apply our results to a geometrically finite, purely loxodromic, Kleinian group F . We construct a subshift S corresponding to (F ) We do not know if S is a SFT. ....

....the action of F on CP; F (viewed as a discrete group of isometries of H 3 ) is geometrically finite; F ) ae [ 2r i=1 D i . Vice versa, assume that F is a finitely generated, free, purely loxodromic Kleinian group. Then F is a Schottky group. See [Mas, X. H] We now recall the results of Bowen [Bow2], who applies the tools of thermodynamics formalism to compute dimH (F ) for a Schottky group F . For convenience we assume that the curves C 1 ; C 2r lie in C. Then dimH (F ) is computed with respect to the Euclidean metric on C. The sets D i (F ) i = 1; 2r; form a Markov ....

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R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S., 50 (1979), 11-26.

....S 1 . Notes and references. The present paper evolved from the preprint [17] written in 1984; many of the results of that preprint appear in x2 and x3 below. Bowen applied the machinery of symbolic dynamics, Markov partitions and Gibbs states to study the Hausdorff dimension of limit sets in [4]. Bodart and Zinsmeister studied H: dimJ(z 2 c) using a Monte Carlo algorithm [3] See also [28] and [23] for calculations for quadratic polynomials. This paper belongs to a three part series. Parts I and II study the continuity of Hausdorff dimension in families of Kleinian groups and ....

R. Bowen. Hausdorff dimension of quasi-circles. Publ. Math. IHES 50(1978), 11--25.

....lim sup n 1 j(f n ) 0 (z)j = 1. The reader may also consult [Mc] for the proof. The equality DD(J(f) hD(J(f) has been shown in Chapter 8 of [PU1] The general scheme of the proof of Theorem 1. 11 is the following: First, it follows easily from the classical Bowen type formula formula (see [Bo1], comp Chapter 6 of [PU1] that hD(J(f) DD(J(f ) A generalization of this formula will be discussed in Section 2 devoted to parabolic and hyperbolic rational functions. Second, using Pesin s theory type and Katok s type considerations (see Chapter [8] of [PU1] one shows that hD(J(f) DD(J(f ....

....caused by the existence of invariant probability measures supported on (periodic) orbits of parabolic points. In case of hyperbolic maps the only zero of P(t) we also denote by s(f ) The following theorem which is classical in the hyperbolic case and called BowenManning McCluskey formula (see [Bo1], comp. MC] has been proved in in the parabolic case in [DU3] It can be considered as a strengthening of Theorem 1.11 and partial answer to the Problem 1.14. Theorem 2.5. If a rational function f : J(f) J(f) is expansive, then we have the following equalities DD(J(f) hD(J(f) ffi(f ) ....

R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES, 50 (1980), 11-25.

....root of the equation P (sOE) 0: 3) For the Moran constructions, s is the Hausdorff dimension of the limit set as well as its box dimension. We show that this is true for general symbolic geometric constructions provided i = i = i . See Corollary 1. The equation (3) was discovered by Bowen [Bo2] and seems to be universal. We will show that all known equations previously used to compute the Hausdorff dimension (for example equation (2) coincide with or are particular cases of (3) We emphasize that for the general symbolic geometric constructions the measure m is an equilibrium ....

R. Bowen, C. Series, Hausdorff Dimension of Quasi-circles, Publ. Math. IHES 50 (1979), 11--25.

....s is the unique root of the equation P (sOE) 0: 3) For the Moran constructions, s is the Hausdorff dimension of the limit set as well as its box dimension. We show that this is true for general symbolic Moran constructions (see Proposition 3 and Theorem 5. Equation (3) was discovered by Bowen [Bo2] and seems to be universal. We will show that all known equations previously used to compute the Hausdorff dimension (for example equation (2) coincide with or are particular cases of (3) For the general symbolic geometric constructions the measure m is an equilibrium measure and admits the ....

R. Bowen and C. Series, Hausdorff Dimension of Quasi-circles, Publ. Math. IHES 50 (1979), 11--25.

....group Gamma = fl 1 ; fl n is a free group contained in PSL(2; C ) for 1991 Mathematics Subject Classification. 58F20, 58F11, 11J70, 11J06. Typeset by A M S T E X 2 O. JENKINSON AND M. POLLICOTT which, in particular, the sets D(fl i ) fz 2 C : jfl 0 i (z)j 1g are disjoint (cf. [Bowen] and [P S] The limit set ae C is the set of accumulation points of the sequence fl i 1 : fl i k (0) say. The value d = dimH has various important interpretations relating to the corresponding surface of negative curvature. For example it equals the topological entropy of the geodesic flow ....

....The particular choice f(x) Gammas log jT 0 (x)j (restricted to E 2 ) gives the following result. Lemma 1. There is a unique solution 0 s 1 to P ( Gammas log jT 0 j) 0, which occurs precisely at s = dimH (E 2 ) Falconer] Bed] Formulae of this general type have a long history. Bowen [Bowen] showed that the dimension of quasi circles could be studied using thermodynamic ideas. Subsequently, Ruelle [Ru2] applied the approach to hyperbolic rational maps and McCluskey Manning [Mc M] studied Smale horseshoes. Our example is covered by the standard theory of cookie cutters (see for ....

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes ' Etudes Sci. Publ. Math. 50 (1979), 11--25.

.... P g (s ) Gamma P g (s ) P (s k ) P (s k ) P g (s ) for every real number s. As is arbitrary, we obtain CP (s k ) P (s k ) P g (s ) The desired statement follows immediately from Proposition 3 and Theorem 2. Bowen s equation was introduced by Bowen in [2]. It was first proven by Ruelle in [8] that the Hausdorff dimension of a repeller of a C 1 ff conformal map is given by Bowen s equation. In [3] Falconer showed that the Hausdorff and box dimensions of such repellers coincide. For general C 1 conformal maps, the statement of Theorem 4 was ....

R. Bowen, Hausdorff dimension of quasi-circles, Inst. Hautes ' Etudes Sci. Publ. Math. 50 (1979), 259--273.

....space and assume that f : X X is a continuous map. Denote by Omega the nonwandering set of f . An interesting and a nontrivial invariant of f is HD( Omega Gamma1375 Hausdorff dimension of Omega Gamma It is usually a highly nontrivial problem to find HD( Omega Gamma3 The seminal work of Bowen [Bow2] gives HD( Omega Gamma as the solution to P (tOE) 0 for some special expanding maps. Here P (g) denotes the topological pressure. See also [Rue2] and the recent works [Bar] and [Fri2] Denote by E the set of all f invariant ergodic probability Borel measures on M . Let HD( 2 E be the ....

....[You] L Y] and [Fri, 1 2] In the above references HD( is given in terms of entropy of f (along a foliation) and the Lyapunov exponents. As the support of lies in Omega it follows that HD( Omega Gamma HD( Hence HD( Omega Gamma sup 2E HD( 0:1) In fact in the examples studied in [Bow2] and [Rue2] one has the equality in (0.1) In these cases HD( Omega Gamma = HD( and is a unique Gibbs measure given by thermodynamics formalism which is equivalent (absolutely continuous) to the Hausdorff measure on Omega Gamma See also [Fri2] In general a strict inequality holds in ....

R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES 50 (1979), 11-25.

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Rufus Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S. 50 (1979), 11--26.

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R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES, 50 (1980), 11-25.

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R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S. 50 (1979), 11--26.

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R. Bowen (1979). Hausdorff dimension of quasi-circles, Publ. Math. I.H.E.S. 50, 11--26.

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