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in this paper we study fractal geometric aspects of the limit set @xmath3 of a normal subgroup @xmath0 of some given non - elementary kleinian group @xmath1 acting on @xmath4-dimensional hyperbolic space @xmath5 . we always assume that @xmath2 contains a coset which is represented by some loxodromic isometry @xmath6 . it is well known that in this situation @xmath3 coincides with the limit set @xmath7 of the larger group @xmath1 . however , a comparison of finer aspects of these two limit sets usually turns out to be far more involved , as can be seen , for instance , in the work of brooks @xcite and rees @xcite . + in this paper we investigate the set @xmath8 of directions at some arbitrary point @xmath9 on the manifold @xmath10 associated with @xmath0 for which the resulting geodesic movement on @xmath10 eventually escapes from every compact region on @xmath10 , but which is nevertheless contained in the @xmath11-neighbourhood of some sequence of closed loops starting and ending at @xmath9 on @xmath10 , for each @xmath12 . that is , we consider the transient limit set @xmath8 of @xmath0 , given by @xmath13 here , @xmath14 refers to the hyperbolic metric in @xmath5 and @xmath15 denotes a @xmath16-parametrisation of the geodesic ray from the origin to @xmath17 . + the following theorem gives the main result of this paper . let @xmath1 and @xmath0 be given as above . we then have @xmath18    this theorem gives a extension of results by fernndez and melin @xcite , who studied the set @xmath19 of escaping , not necessarily loop - approximable directions on a complete oriented non - compact riemann surface @xmath20 with fundamental group @xmath21 . that is , @xmath22 , where @xmath23 refers to the boundary at infinity of @xmath24 and @xmath25 denotes the hyperbolic metric in @xmath24 . the main result of ( * ? ? ? * theorem  1 ) was to establish the following tricothomy .    * if @xmath20 has finite area , then @xmath19 is countable . * if brownian motion on @xmath20 is transient , then @xmath19 has full lebesgue measure . * if @xmath20 has infinite area and brownian motion is recurrent , then @xmath19 has zero lebesgue measure , but its hausdorff dimension is equal to @xmath26 . therefore , for hyperbolic manifolds which are normal coverings of some hyperbolic manifold and which posess a loxodromic representative @xmath27 , our main theorem extends the results by fernndez and melin to arbitrary dimensions and to the situation where the boundary at infinity of hyperbolic space is replaced by the limit set of the fundamental group . however , let us emphasize that our proof does require the existence of a loxodromic @xmath27 , and hence our extension is restricted to normal coverings with this property . our proof hinges on two renormalisation procedures . these are used to locate a certain family of subsystems within the loop - approximable , non - recurrent dynamics on the manifold @xmath28 . at the boundary of the universal covering space these subsystems are described by a family of cantor sets contained in @xmath3 , and here the key observation is that this family contains cantor sets whose hausdorff dimension is arbitrarily close to the hausdorff dimension of @xmath7 . these cantor sets are constructed inductively using the following two renormalisation procedures . the first of these employs a well - known construction by bishop and jones for computing the hausdorff dimension of bounded dynamics ( see @xcite , @xcite ) . this construction gives rise to a certain weighted scaling law , which we then apply a sufficient number of times in order to prepare for the second renormalisation step . on the manifold @xmath28 this first step of the overall construction corresponds to a family of well separated quasi - geodesics within a bounded region of @xmath28 , each starting at the same point . the second renormalisation procedure consists in prolonging each of these quasi - geodesics by a long geodesic segment which is contained in the projection of the axis @xmath29 of @xmath30 onto @xmath28 , and which is chosen such that it leads out of the bounded region which contained the original quasi - geodesics . the resulting dynamical behaviour on @xmath28 is sketched in figure  [ mnf ] . we assume that the reader is familiar with the proof of bishop and jones result on the relationship between the exponent of convergence @xmath31 of a non - elementary kleinian group @xmath1 and the hausdorff dimension of the radial limit set @xmath32 of @xmath1 ( see @xcite and for a more detailed proof @xcite ) . here , the reader might like to recall that the radial limit set @xmath32 represents those limit points @xmath17 for which the projection of a hyperbolic ray towards @xmath17 returns infinitely often to some compact part of the manifold associated to @xmath1 . @xmath33    .,title="fig : " ] [ mnf ] throughout , let @xmath1 be a non - elementary kleinian group acting on @xmath4-dimensional hyperbolic space @xmath5 . also , let @xmath0 be a normal subgroup of @xmath1 such that @xmath2 contains a coset @xmath34 $ ] , for some loxodromic @xmath6 . moreover , we always assume without loss of generality that @xmath35 is an element of the axis @xmath36 of @xmath30 , and we let @xmath37 ( @xmath38 resp . ) refer to the repulsive ( attractive resp . ) fixed point of @xmath30 . next , recall that to any arbitrary kleinian group @xmath21 we can associate its truncated poincar series @xmath39 , as well as its poincar series @xmath40 . these series are given , for @xmath41 and @xmath42 , by @xmath43 the abzissa of convergence of the infinite series @xmath40 is called the exponent of convergence of @xmath21 , and it will be denoted by @xmath44 . here , note that bishop and jones @xcite ( see also @xcite ) showed that if @xmath21 is non - elementary , then we always have that @xmath45 also , we will make use of the following standard facts and notations for the poincar model @xmath46 of the @xmath4-dimensional hyperbolic space . let @xmath47 refer to the hyperbolic ball centred at @xmath48 of radius @xmath49 , and let @xmath50 denote the radial projection from the origin to the boundary @xmath51 of hyperbolic space . that is , for @xmath52 we have @xmath53 , where @xmath54 refers to the euclidean straight line between the origin and @xmath17 . also , we require the following analogue of pythagoras s theorem for hyperbolic triangles . for this , consider a hyperbolic triangle with sides of finite lengths @xmath55 , @xmath56 and @xmath57 , and with @xmath58 denoting the angle opposite to the side of length @xmath55 . a straightforward application of the hyperbolic cosine rule ( see e.g. @xcite ) then gives that there exists a constant @xmath59 , depending only on @xmath60 , such that @xmath61 moreover , we require the following additional observation from elementary hyperbolic geometry . for this , consider some arbitrary hyperbolic geodesic @xmath62 which does not contain the origin , and let @xmath63 denote the geodesic ray connecting the origin with one of the endpoints @xmath64 of @xmath65 . also , let @xmath66 refer to the summit of the geodesic @xmath65 . that is , @xmath66 is uniquely determined by @xmath67 . a straightforward exercise in hyperbolic geometry then shows that there exists a universal constant @xmath68 such that @xmath69 in fact , an elementary calculation shows that @xmath70 is equal to @xmath71 , which is often referred to as schweikart s constant .    for a sequence @xmath72 of distinct points in @xmath5 , let @xmath73\ ] ] denote the quasi - geodesic path obtained by connecting @xmath74 and @xmath75 with the unique geodesic arc between them , for each @xmath76 . a standard observation from hyperbolic geometry then shows that if the lengths of these geodesic segments are uniformly bounded away from @xmath77 , and if each of the angles between adjacent geodesic segments is uniformly bounded from below by some @xmath78 , then @xmath79 $ ] is a quasi - geodesic ray towards a unique point at infinity . that is , each @xmath74 is , with respect to the hyperbolic metric , uniformly bounded ( depending on @xmath60 ) away from the geodesic ray from @xmath80 towards the uniquely determined limit at the boundary at infinity of the sequence @xmath81 . finally , note that we use the common notation @xmath82 if two sequences of positive real numbers @xmath83 and @xmath84 are comparable , that is , if the ratio @xmath85 is uniformly bounded from below by @xmath86 and from above by @xmath57 , for some @xmath87 and for all @xmath76 . let us begin with by giving our first renormalisation procedure . here , a _ geodesic @xmath0-tree @xmath88 rooted at @xmath9 _ refers to an infinite tree whose set of vertices @xmath89 is contained in @xmath90 and whose edges are finite geodesic segments between the vertices , such that each vertex @xmath91 has a finite set @xmath92 of successors of cardinality at least @xmath93 and such that each element in @xmath94 has a unique predecessor .    in the following , let @xmath95 be defined by @xmath96 , for each @xmath97 . note that our first renormalisation procedure is well known for the special case in which each @xmath95 lies in the orbit @xmath98 . in this situation , its outcome has already been obtained in ( * ? ? ? * proposition 3.5 ) . the novelty here is that for the normal subgroup @xmath0 the result of ( * ? ? ? * proposition 3.5 ) continues to hold for each element of the orbit @xmath99 of the origin under @xmath100 .    * recurrent renormalisation procedure ( [ rrp ] ) . * @xmath101    _ for each @xmath102 , there exist @xmath103 , @xmath104 and @xmath105 such that for each @xmath106 and @xmath107 there exists a geodesic @xmath0-tree @xmath108 rooted at @xmath109 with the following properties . _    * if @xmath110 , then @xmath111 for each @xmath112 . * if @xmath112 for some @xmath110 , then @xmath113 . * if @xmath114 for some @xmath110 , then @xmath115 and @xmath116 . * for each @xmath110 we have @xmath117    we will say that the so derived family @xmath118 is obtained by applying the recurrent renormalisation procedure to @xmath110 . as already mentioned before , for @xmath119 the assertion in this procedure has been obtained in ( * ? ? ? * proposition 3.5 ) ( see also @xcite ) , and we refer to these papers for the proof in this case . in fact , note that the main idea of the proof of ( * ? ? ? * proposition  3.5 ) consists of a geometrization of the rate of increase , for @xmath120 tending to infinity , of the truncated poincar series @xmath121 for @xmath122 . for the proof of the general situation , that is , for some arbitrary @xmath107 , note that the value of the truncated poincar series associated with @xmath0 does not change if we exchange the observation point @xmath123 by some arbitrary point in @xmath124 . more precisely , since @xmath0 is normal in @xmath1 , we have for each @xmath97 and @xmath125 , @xmath126 using this observation , the assertion now follows from a straightforward adaptation of the arguments in the proof of ( * ? ? ? * proposition 3.5 ) .    * transient renormalisation procedure ( [ trp ] ) . * @xmath101     _ for @xmath127 , @xmath107 and @xmath128 , let @xmath108 denote the geodesic @xmath0-tree obtained in the recurrent renormalisation procedure _ [ rrp]_. then there exists a constant @xmath129 such that for each @xmath130 sufficiently large and for each @xmath106 with @xmath131 , we have that @xmath132 where @xmath133 . we will say that the ball @xmath134 is obtained by starting at @xmath109 and applying the transient renormalisation procedure @xmath135 times . _    @xmath33    .,title="fig : " ] [ etaplus ]    let @xmath108 and @xmath106 be given as stated in the renormalisation procedure . let us first show that @xmath136 lies always close to the summit @xmath137 of the geodesic @xmath138 , the image of the axis @xmath139 under @xmath140 . indeed , since @xmath141 , the statement in ( 1 ) of [ rrp ] implies that there exists @xmath142 such that @xmath143 ( see also figure  [ etaplus ] ) . the elementary observation in ( [ axis ] ) shows that the distance from the summit of a geodesic to each of the two rays from the origin to the endpoints of the geodesic is less than @xmath70 . thus , by construction , we have that @xmath144    recall that in the recurrent renormalisation procedure [ rrp ] we have already derived the existence of the parameter @xmath145 , which is the upper bound of the lengths of the edges in the tree @xmath146 . next , consider the geodesic @xmath147 containing the points @xmath148 and @xmath109 ; one of its endpoints will be @xmath149 . assume , by way of contradiction , that the distance between @xmath109 and the summit @xmath137 of @xmath147 is larger than @xmath150 . projecting onto the manifold @xmath28 associated to @xmath0 and using the hyperbolic triangle inequality , we obtain a contradiction to the fact that @xmath151 . it immediately follows that @xmath152 where the comparability constant depends only on the distance @xmath150 , and therefore , only on @xmath0 and @xmath153 . the statement now follows by applying @xmath154 to the ball @xmath155 , which immediately gives that @xmath156 where @xmath157 . let us remark that the constant @xmath158 in [ rrp ] and the constant @xmath70 in [ trp ] are independent of each other . also , the statements in [ rrp ] continue to hold if we replace @xmath158 by a smaller positive number , and the same holds for @xmath70 in [ trp ] . therefore , for the remainder of this paper , when applying [ rrp ] and [ trp ] , we use @xmath159 instead of @xmath158 and @xmath70 . let us first observe that it is sufficient to prove the assertion in the theorem for the case in which @xmath160 indeed , this can immediately be seen by way of contradiction as follows . suppose that @xmath161 . since diminishing a set by a subset of smaller hausdorff dimension does not alter the hausdorff dimension of that set and using the well known fact that @xmath162 ( see @xcite and @xcite ) , we obtain @xmath163 therefore , we can now assume , without loss of generality , that @xmath164 the rough strategy for proving the main theorem in this case is as follows . for some arbitrary given @xmath165 , we construct a certain cantor set @xmath166 , and then show that @xmath167 . by the arbitrary choice of @xmath153 , the theorem then follows . the idea of the cantor set construction is to start at the origin and then to perform an alternating inductive process using both renormalisation procedures . the building block of this process is that we first apply the recurrent renormalisation procedure [ rrp ] sufficiently many times until the resulting power of @xmath168 is large enough ( in fact , this number of times depends on the outcome of the step to come ) . after that , we perform the transient renormalisation procedure [ trp ] sufficiently many times , without loosing the control on the distortion ( in particular , this step will guarantee that out cantor set contains only transient limit points ) . more precisely , let @xmath169 be given . then @xmath153 determines the width @xmath145 of the recurrent renormalisation procedure [ rrp ] . having fixed @xmath145 , we choose @xmath170 , the number of times we are going to apply the transient renormalisation procedure [ trp ] , so that @xmath171 this choice of @xmath135 will guarantee that the cantor set @xmath172 we are going to construct will be contained in @xmath173 . finally , we choose @xmath174 to be minimal with respect to the property @xmath175 let us now come to the explicit construction of @xmath172 . as already mentioned , the construction starts at the origin , and we set @xmath176 . then , the first step is to apply the recurrent renormalisation procedure [ rrp ] @xmath177 times , starting at @xmath178 . according to [ rrp ] , this gives rise to a set of hyperbolic balls @xmath179 whose radial projections to the boundary @xmath51 are pairwise disjoint and of comparable diameter . the set of centres of these balls in @xmath5 will be denoted by @xmath180 . then , the second step is to apply the transient renormalisation procedure [ trp ] @xmath135 times to each element in @xmath180 . the set of centres of the so obtained hyperbolic balls will be denoted by @xmath181 . this represents the start of the induction , and we then continue as follows . assume that the sets @xmath182 and @xmath183 have been constructed , for some @xmath76 . to each of the points in @xmath183 we then apply the recurrent renormalisation procedure [ rrp ] @xmath177 times . the set of centres of these so obtained hyperbolic balls gives the set @xmath184 . next , we apply the transient renormalisation procedure [ trp ] @xmath135 times to each of the elements in @xmath184 . the set of centres of these so obtained hyperbolic balls gives rise to the set @xmath185 . this finishes our alternating inductive argument ( see also figure  [ induct ] ) , and we can now use it to define our desired cantor set @xmath172 by @xmath186 here , @xmath187 refers to the constant which we specified at the end of section [ 3 ] . + next , observe that in this cantor set construction we have good control over the distortion , when going from one generation in the construction to the next . that is , by using [ rrp ] @xmath188 , [ trp ] and the condition in ( [ constants ] ) , we have the following crucial estimate , for each @xmath76 , @xmath189 using a straightforward generalisation of the folklore arguments from fractal geometry of ( * ? ? ? * lemma 2.5 ) and ( * ? ? ? * corollary 2.6 ) , the latter estimate immediately gives that @xmath190 it remains to show that the set @xmath172 is contained in @xmath8 . for this , note that , by viewing the construction of @xmath172 from within @xmath5 , the set @xmath172 gives rise to a geodesic @xmath1-tree which is rooted at the origin and whose vertex set is equal to @xmath191 . by construction , this tree has the property that the lengths of the constituting geodesic edges and the angles formed by adjacent edges are uniformly bounded away from zero . therefore , each path in this tree starting at the origin is a quasi - geodesic heading towards a uniquely determined point at infinity . clearly , the projection of each of these quasi - geodesics onto the manifold @xmath10 gives some piecewise geodesic movement in @xmath10 which has the following properties . if an edge in the tree starts at a vertex in @xmath192 and ends at a point in @xmath182 , then in @xmath10 this edge is represented by a geodesic loop of hyperbolic length at most @xmath193 . obviously , this loop must then be contained in a bounded region of @xmath10 of diameter at most @xmath193 . whereas , if an edge in the tree starts at a vertex in @xmath182 and ends at a point in @xmath183 , using [ trp ] , we then have that in @xmath10 this edge represents a geodesic segment in @xmath10 which starts in the previous bounded region and then heads straight towards the end of @xmath10 associated with the attractive fixed point @xmath38 of @xmath30 . moreover , the condition in ( [ transient ] ) guarantees that the hyperbolic length of that segment is at least equal to @xmath194 , and this shows that it s end point is separated by at least @xmath195 from the previous bounded region . @xmath196              m. rees , checking ergodicity of some geodesic flows with infinite gibbs measure , _ ergod . th . @xmath197 dynam . * 1 * ( 1981 ) , 107133 . m. rees , divergence type of some subgroups of finitely generated fuchsian groups , _ ergod . @xmath197 dynam . sys . _ * 1 * ( 1981 ) , 209221 . b. o. stratmann , the exponent of convergence of kleinian groups ; on a theorem of bishop and jones , _ progr . in probab . _ * 57 * ( 2004 ) , 93107 .
the main result of this paper is to show that if @xmath0 is a normal subgroup of a kleinian group @xmath1 such that @xmath2 contains a coset which is represented by some loxodromic element , then the hausdorff dimension of the transient limit set of @xmath0 coincides with the hausdorff dimension of the limit set of @xmath1 . 
 this observation extends previous results by fernndez and melin for riemann surfaces .    * 
 ams classification : * 30 f 40 , 37 f 99 , 37 f 30 , 28 a 80 .    * 
 keywords : * kleinian groups ; poincar exponent ; fractal geometry ; dissipative dynamics .