document: recall a polynomial @xmath0 is _ postcritically finite _ if the forward orbits of its critical points are finite . 
   then the filled julia set of @xmath1 contains a forward invariant , finite topological tree , called the _ hubbard tree _ @xcite . 
   the _ core entropy _ of @xmath1 is the topological entropy of the restriction of @xmath1 to its hubbard tree . 
   we shall restrict ourselves to quadratic polynomials . 
   given @xmath2 , the external ray at angle @xmath3 determines a postcritically finite parameter @xmath4 in the mandelbrot set @xcite . 
   we define @xmath5 to be the core entropy of @xmath6 . 
   the main goal of this paper is to prove the following result :    [ t : main ] the core entropy function @xmath7 extends to a continuous function from @xmath8 to @xmath9 . 
   the theorem answers a question of w. thurston , who first introduced and explored the core entropy of polynomials . 
   as thurston showed , the core entropy function can be defined purely combinatorially , but it displays a rich fractal structure ( figure [ f : core ] ) , which reflects the underlying geometry of the mandelbrot set . 
   [ f : core ]    the concept of core entropy generalizes to complex polynomials the entropy theory of real quadratic maps , whose monotonicity and continuity go back to milnor and thurston @xcite : indeed , the invariant real segment is replaced by the invariant tree , which captures all the essential dynamics . 
   on the topological side , hubbard trees have been introduced to classify postcritically finite maps @xcite , @xcite , and their entropy provides a new tool to study the parameter space of polynomials : for instance , the restriction of @xmath5 to non - real veins of the mandelbrot set is also monotone @xcite , which implies that the lamination for the mandelbrot set can be reconstructed by looking at level sets of @xmath10 . note by comparison that the entropy of @xmath11 on its julia set is constant , independent of @xmath3 , hence it does not give information on the parameter . 
   furthermore , the value @xmath5 equals , up to a constant factor , the hausdorff dimension of the set of biaccessible angles for @xmath11 ( @xcite , @xcite ) . for dendritic julia sets , the core entropy also equals the asymptotic stretch factor of @xmath11 as a rational map @xcite . 
   a rational external angle @xmath3 determines a postcritically finite map @xmath11 , which has a finite hubbard tree @xmath12 . 
   the simplest way to compute its entropy is to compute the markov transition matrix for the map @xmath11 acting on @xmath12 , and take ( the logarithm of ) its leading eigenvalue : however , this requires to know the topology of @xmath12 , which changes wildly after small perturbations of @xmath3 .    in this paper , we shall by - pass this issue by leveraging an algorithm , devised by thurston , which considers instead a larger matrix , whose entries are _ pairs _ of postcritical angles ( see section [ s : algo ] ) ; no knowledge of the topology of the hubbard tree is required . in order to prove @xmath10 is continuous , we develop an infinite version of such algorithm , which is defined for _ any _ angle @xmath13 . 
   in particular , instead of taking the leading eigenvalue of a transition matrix , we shall encode the possible transitions in a directed graph , which will now have countably many vertices . by taking the _ spectral determinant _ of such graph ( see section [ s : spectral ] for a definition ) , for each angle @xmath13 , we construct a power series @xmath14 which converges in the unit disk and such that its  smallest zero " @xmath15 is related to the core entropy by @xmath16 we then produce an algorithm to compute each coefficient of the taylor expansion of @xmath14 , and show that essentially these coefficients vary continuously with @xmath3 : the result follows by rouch s theorem .    as a corollary of our method 
   , we shall prove that the entropy function is actually hlder continuous at angles @xmath3 such that @xmath17 ( using renormalization , it can be proven that @xmath10 is not hlder continuous where @xmath18 ) .      on the way to our proof 
   , we shall develop a few general combinatorial tools to deal with growth rates of countable graphs , which may be of independent interest . 
   we define a countable graph @xmath19 with bounded ( outgoing ) degree to have _ bounded cycles _ if it has finitely many closed paths of any given length . in this case 
   , we define the _ growth rate _ of @xmath19 as the growth rate with @xmath20 of the number of closed paths of length @xmath20 ( see definition [ d : growth ] ) ; as it turns out ( theorem [ t : rootofp ] ) , the inverse of the growth rate equals the smallest zero of the following function @xmath21 , constructed by counting the multi - cycles in the graph : @xmath22 where @xmath23 is the number of components of the multi - cycle @xmath24 , and @xmath25 is its length . 
   ( for the definition of multi - cycle , see section [ s : graphs ] ; note that our graphs have finite outgoing degree but possibly infinite ingoing degree , hence the adjacency operator has infinite @xmath26-norm and the usual spectral theory ( see e.g. @xcite ) does not apply ) . 
   we then define a general combinatorial object , called _ labeled wedge _ , which consists of pairs of integers which can be labeled either as being _ separated _ ( representing two elements of the postcritical set which lie on opposite sides of the critical point ) or _ non - separated _ ; to such object we associate an infinite graph , and prove that it has bounded cycles , hence one can apply the theory developed in the first part ( see theorem [ t : zero ] ) . 
   finally , we shall apply these combinatorial techniques to the core entropy ( section [ s : core ] ) ; indeed , we associate to any external angle @xmath3 a labeled wedge @xmath27 , hence an infinite graph @xmath28 , and verify that :    1 . 
   the growth rate of @xmath28 varies continuously as a function of @xmath3 ( theorem [ t : newmain ] ) ; 2 . 
   the growth rate of @xmath28 coincides with the core entropy for rational angles ( theorem [ t : coincide ] ) . 
   let us remark that , as a consequence of monotonicity along veins and theorem [ t : main ] , the continuous extension we define also coincides with the core entropy for parameters which are not necessarily postcritically finite , but for which the julia set is locally connected and the hubbard tree is topologically finite .      the core entropy of polynomials has been introduced by w. thurston around 2011 ( even though there are earlier related results , e.g. @xcite , @xcite , @xcite ) but most of the theory is yet unpublished ( with the exception of section 6 in @xcite ) . 
   several people are now collecting his writings and correspondence into a foundational paper @xcite . 
   in particular , the validity of thurston s algorithm has been proven by tan l. , gao y. , and w. jung ( see @xcite , @xcite ) . 
   continuity of the core entropy along principal veins in the mandelbrot set is proven by the author in @xcite , and along all veins by w. jung @xcite . 
   note that the previous methods used for veins do not easily generalize , since the topology of the tree is constant along veins but not globally . 
   after @xcite , alternative proofs of monotonicity and continuity of entropy for real maps are given in @xcite , @xcite ( the present proof independently yields continuity ) . 
   biaccessible external angles and their dimension have been discussed in @xcite , @xcite , @xcite , @xcite , @xcite . 
   the method we use to count closed paths in the graph bears many similarities with the theory of _ dynamical zeta functions _ 
   @xcite , and several forms of the spectral determinant are used in thermodynamic formalism ( see e.g. @xcite , @xcite ) . moreover , the spectral determinant @xmath21 we use is an infinite version of the _ clique polynomial _ used in @xcite to study finite directed graphs with small entropy .      the motivation for our combinatorial construction is thurston s algorithm to compute the core entropy for rational angles , which we shall now describe . 
   let @xmath2 a rational number @xmath29 . 
   then the external ray at angle @xmath3 in the mandelbrot set lands at a misiurewicz parameter , or at the root of a hyperbolic component : let @xmath1 denote the corresponding postcritically finite quadratic map . we shall call @xmath30 the critical point of @xmath1 , and for each @xmath31 , @xmath32 the @xmath33 iterate of the critical point . 
   recall that the _ hubbard tree _ of @xmath1 is the union of the regulated arcs @xmath34 $ ] for all @xmath35 ( for more details , see @xcite ) . in the postcritically finite case , it is a finite topological tree , and it is forward - invariant under @xmath1 ( see figure [ f : tree ] ) . 
   it is possible to compute the core entropy of @xmath1 by writing the markov transition matrix for the action of @xmath1 on the tree , and take the logarithm of its leading eigenvalue . 
   however , given the external angle @xmath3 it is quite complicated to figure out the topology of the tree : the following algorithm by - passes this issue by looking at pairs of external angles .    in order to explain the algorithm in more detail , 
   let us remark that a rational angle @xmath3 is eventually periodic under the doubling map @xmath36 ; that is , there exist integers @xmath37 and @xmath38 such that the elements of the set @xmath39 are all distinct modulo @xmath40 , and @xmath41 . 
   the elements of @xmath42 will be called _ postcritical angles _ ; the number @xmath43 is called the _ period _ of @xmath3 , and @xmath44 is the _ pre - period_. if @xmath45 , we shall call @xmath3 _ 
   purely periodic_. denote @xmath46 the partition of the circle @xmath47 in the two intervals @xmath48 moreover , for each @xmath49 let us denote @xmath50 , which we see as a point in @xmath51 . 
   let us now construct a matrix @xmath52 which will be used to compute the core entropy . denote @xmath53 the set of unordered pairs of ( distinct ) elements of @xmath42 : this is a finite set , any element of which will be denoted @xmath54 . 
   we now define a linear map @xmath55 by defining it on its basis vectors in the following way :    * if @xmath56 and @xmath57 belong to the same element of the partition @xmath46 , or at least one of them lies on the boundary , we shall say that the pair @xmath54 is _ non - separated _ , and define @xmath58 * if @xmath56 and @xmath57 belong to the interiors of two different elements of @xmath46 , then we say that @xmath54 is _ separated _ , and define @xmath59 ( in order to make sure the formulas are defined in all cases , we shall set @xmath60 whenever @xmath61 ) .    by abuse of notation , we shall also denote @xmath52 the matrix representing the linear map @xmath52 in the standard basis . as suggested by thurston in his correspondence , 
   the leading eigenvalue of @xmath52 gives the core entropy :    [ t : algo ] let @xmath3 a rational angle.then the core entropy @xmath5 of the quadratic polynomial of external angle @xmath3 is related to the largest real eigenvalue @xmath62 of the matrix @xmath52 by the formula @xmath63    the explanation of this algorithm is the following . 
   let @xmath64 be a complete graph whose vertices are labeled by elements of @xmath42 , and which we shall consider as a topological space . 
   ( more concretely , we can take the unit disk and draw segments between all possible pairs of postcritical angles on the unit circle : the union of all such segments is a model for @xmath64 ( see figure [ f : chords_algo ] ) ) . 
   let us denote by @xmath65 the hubbard tree of @xmath1 . 
   we can define a continuous map @xmath66 which sends the edge with vertices @xmath56 , @xmath57 homeomorphically to the regulated arc @xmath34 $ ] in @xmath65 ( except in the case @xmath67 , where we map all the edge to a single point ) . finally , we can lift the dynamics @xmath68 to a map @xmath69 such that @xmath70 . in order to do so , 
   note that :    * if the pair of angles @xmath54 is non - separated , then @xmath1 maps the arc @xmath34 $ ] homeomorphically onto @xmath71 $ ] ; hence to define @xmath72 we have to lift @xmath1 so that it maps the edge @xmath73 $ ] homeomorphically onto @xmath74 $ ] ; * if instead @xmath54 is separated , then the critical point @xmath30 lies on the regulated arc @xmath34 $ ] in the hubbard tree ; thus , @xmath1 maps the arc @xmath75 = [ c_i , c_0 ] \cup [ c_0 , c_j]$ ] onto @xmath76 \cup [ c_1 , c_{j+1}]$ ] ; so we define @xmath72 by lifting @xmath1 and so that it maps @xmath73 $ ] continuously onto the union @xmath77 \cup [ x_1 , x_{j+1}]$ ] . 
   the map @xmath72 is a markov map of a topological graph , hence its entropy is the logarithm of the leading eigenvalue of its transition matrix , which is by construction the matrix @xmath52 . in case @xmath3 
   is purely periodic , one can prove that the map @xmath78 is surjective and finite - to - one , and semiconjugates the dynamics @xmath72 on @xmath64 to the dynamics @xmath1 on @xmath65 , hence @xmath79 more care is needed in the pre - periodic case , since @xmath78 can collapse arcs to points ( for details , see @xcite , @xcite , or @xcite ) . 
   in the following , by _ graph _ we mean a directed graph @xmath19 , i.e. a set @xmath80 of vertices ( which will be finite or countable ) and a set @xmath81 of edges , such that each edge @xmath82 has a well - defined _ source _ @xmath83 and a _ target _ @xmath84 
   ( thus , we allow edges between a vertex and itself , and multiple edges between two vertices ) . given a vertex @xmath85 , 
   the set @xmath86 of its _ outgoing edges _ is the set of edges with source @xmath85 . 
   the _ outgoing degree _ of @xmath85 is the cardinality of @xmath86 ; a graph is _ locally finite _ if the outgoing degree of all its vertices is finite , and has _ bounded outgoing degree _ if there is a uniform upper bound @xmath87 on the outgoing degree of all its vertices . 
   note that we do _ not _ require that the ingoing degree is finite , and indeed in our application we will encounter graphs with vertices having countably many ingoing edges .    we denote as @xmath88 the set of vertices of @xmath19 , and as @xmath89 its set of edges . 
   moreover , we denote as @xmath90 the number of edges from vertex @xmath85 to vertex @xmath91 . 
   a _ path _ in the graph based at a vertex @xmath85 is a sequence @xmath92 of edges such that @xmath93 , and @xmath94 for @xmath95 . 
   the _ length _ of the path is the number @xmath20 of edges , and the set of vertices @xmath96 visited by the path is called its _ vertex - suppoert _ , or just _ support _ for simplicity . 
   similarly , a _ closed path _ based at @xmath85 is a path @xmath92 such that @xmath97 . 
   note that in this definition a closed path can intersect itself , i.e. two of the sources of the @xmath98 can be the same ; moreover , closed paths with different starting vertices will be considered to be different .    on the other hand , a _ 
   simple cycle _ is a closed path which does not self - intersect , modulo cyclical equivalence : that is , a simple cycle is a closed path @xmath92 such that @xmath99 for @xmath100 , and two such paths are considered the same simple cycle if the edges are cyclically permuted , i.e. @xmath92 and @xmath101 designate the same simple cycle . 
   finally , a _ 
   multi - cycle _ is the union of finitely many simple cycles with pairwise disjoint ( vertex-)supports . the length of a multi - cycle is the sum of the lengths of its components .    given a countable graph with bounded outgoing degree , we define the _ adjacency operator _ 
   @xmath102 on the space of summable sequences indexed by the vertex set @xmath80 . 
   in fact , for each vertex @xmath103 we can consider the sequence @xmath104 which is @xmath40 at position @xmath49 and @xmath105 otherwise , and define for each @xmath106 the @xmath107 component of the vector @xmath108 to be @xmath109 equal to the number of edges from @xmath49 to @xmath110 . 
   since the graph has bounded outgoing degree , the above definition can actually be extended to all @xmath111 , and the operator norm of @xmath112 induced by the @xmath113-norm is bounded above by the outgoing degree @xmath114 of the graph . 
   note moreover that for each pair @xmath115 of vertices and each @xmath20 , the coefficient @xmath116 equals the number of paths of length @xmath20 from @xmath49 to @xmath110 . 
   we say a countable graph @xmath19 has _ bounded cycles _ if it has bounded outgoing degree and for each positive integer @xmath20 , @xmath19 has at most finitely many simple cycles of length @xmath20 .    note that , if @xmath19 has bounded cycles , then for each @xmath20 it has also a finite number of closed paths of length @xmath20 , since the support of any closed path of length @xmath20 is contained in the union of the supports of simple cycles with length @xmath117 . 
   thus , for such graphs we shall denote @xmath118 the number of closed paths of length @xmath20 . 
   note that in this case the trace @xmath119 is also well - defined for each @xmath20 , and equal to @xmath120 . 
   [ d : growth ] if @xmath19 is a graph with bounded cycles , we define the _ growth rate _ @xmath121 as the exponential growth rate of the number of its closed paths : that is , @xmath122{c(\gamma , n)}.\ ] ]      if @xmath19 is a finite graph with adjacency matrix @xmath112 , then is well - defined its characteristic polynomial @xmath123 , whose roots are the eigenvalues of @xmath112 . in the following 
   we shall work with the related polynomial @xmath124 which we call the _ spectral determinant _ of @xmath19 . 
   we shall now extend the theory to countable graphs with bounded cycles . 
   it is known that the spectral determinant @xmath21 of a finite graph @xmath19 is related to its multi - cycles by the following formula ( see e.g. @xcite ) : @xmath125 where @xmath25 denotes the length of the multi - cycle @xmath24 , while @xmath23 is the number of connected components of @xmath24 . 
   if @xmath19 is now a ( directed ) graph with countably many vertices and bounded cycles , then the number of multi - cycles of any given length is finite , hence the formula above is still defined as a formal power series . 
   note that we include also the empty cycle , which has zero components and zero length , hence @xmath21 begins with the constant term @xmath40 . 
   now , let @xmath126 denote the number of multi - cycles of length @xmath20 in @xmath19 , and let us define @xmath127{k(\gamma , n)}\ ] ] its growth rate . 
   then the main result of this section is the following :    [ t : rootofp ] suppose we have @xmath128 ; then the formula defines a holomorphic function @xmath129 in the unit disk @xmath130 , and moreover the function @xmath129 is non - zero in the disk @xmath131 ; if @xmath132 , we also have @xmath133 . 
   the proof uses in a crucial way the following combinatorial statement . 
   [ l:1/pt ] let @xmath19 be a countable graph with bounded cycles , and @xmath112 its adjacency operator . 
   then we have the equalities of formal power series @xmath134 where @xmath21 is the spectral determinant . 
   note that , since @xmath21 is a power series with constant term @xmath40 , then @xmath135 is indeed a well - defined power series . 
   the second equality is just obtained by expanding the exponential function in power series . to prove the first equality , 
   let us first suppose @xmath19 is a finite graph with @xmath20 vertices , and let @xmath136 be the eigenvalues of its adjacency matrix ( counted with algebraic multiplicity ) . 
   then @xmath137 hence @xmath138 hence the claim follows since @xmath139 . 
   now , let @xmath19 be infinite with bounded cycles , and let @xmath140 . 
   note that both sides of the equation depend , modulo @xmath141 , only on multi - cycles of length @xmath142 , which by the bounded cycle condition are supported on a finite subgraph @xmath143 . 
   thus by applying the previous proof to @xmath143 we obtain equality modulo @xmath141 , and since this holds for any @xmath144 the claim is proven .    by the root test 
   , the radius of convergence of @xmath145 is @xmath146 . 
   thus , since the exponential function has infinite radius of convergence , the radius of convergence of @xmath147 is at least @xmath146 , and since @xmath148 where @xmath149 has positive coefficients , then the radius of convergence of @xmath147 is exactly equal to @xmath146 . 
   hence , by lemma [ l:1/pt ] , the radius of convergence of @xmath135 around @xmath150 is also @xmath146 . on the other hand , since @xmath128 , then by the root test the power series @xmath21 converges inside the unit disk , and defines a holomorphic function ; thus , @xmath135 is meromorphic for @xmath151 , and holomorphic for @xmath152 , hence @xmath153 if @xmath152 . moreover , 
   if @xmath154 , then the radius of convergence of @xmath135 equals the smallest modulus of one of its poles , hence @xmath146 is the smallest modulus of a zero of @xmath21 . 
   finally , since @xmath135 has all its taylor coefficients real and nonnegative , then the smallest modulus of its poles must also be a pole , so @xmath133 . 
   [ l : perroneig ] if @xmath19 is a finite graph , then its growth rate equals the largest real eigenvalue of its adjacency matrix . 
   note that , by the perron - frobenius theorem , since the adjacency matrix is non - negative , it has at least one real eigenvalue whose modulus is at least as large as the modulus of any other eigenvalue . moreover , if @xmath62 is the largest real eigenvalue , then @xmath155 is the smallest root of the spectral determinant @xmath21 , hence the claim follows from theorem [ t : rootofp ] . 
   let @xmath156 two ( locally finite ) graphs . 
   a _ graph map _ from @xmath157 to @xmath158 is a map @xmath159 on the vertex sets and a map on edges @xmath160 which is compatible , in the sense that if the edge @xmath82 connects @xmath85 to @xmath91 in @xmath157 , then the edge @xmath161 connects @xmath162 to @xmath163 in @xmath158 . 
   we shall usually denote such a map as @xmath164 . 
   weak cover _ of graphs is a graph map @xmath164 such that :    * the map @xmath159 between the vertex sets is surjective ; * the induced map @xmath165 between outgoing edges is a bijection for each @xmath166 . 
   note that the map between outgoing edges is defined because @xmath167 is a graph map , and @xmath167 also induces a map from paths in @xmath157 to paths in @xmath158 . as a consequence of the definition of weak cover , you have the following _ unique path lifting _ property :    let @xmath164 a weak cover of graphs . then given @xmath166 and @xmath168 , for every path @xmath24 in @xmath158 based at @xmath91 
   there is a unique path @xmath169 in @xmath157 based at @xmath85 such that @xmath170 . 
   let @xmath166 and @xmath168 , and let @xmath171 be a path in @xmath158 based at @xmath91 . since the map @xmath172 is a bijection , there exists exactly one edge @xmath173 such that @xmath174 . by compatibility , the target @xmath175 of @xmath176 projects to the target of @xmath177 , 
   hence we can apply the same reasoning and lift the second edge @xmath178 uniquely starting from @xmath175 , and so on . 
   an immediate consequence of the property is the following    [ l : easyineq ] each graph map @xmath164 induces a map @xmath179 for each @xmath180 and each @xmath181 . 
   moreveor , if @xmath167 is a weak cover , then @xmath182 is injective . 
   a general way to construct weak covers of graphs is the following . 
   suppose we have an equivalence relation @xmath183 on the vertex set @xmath80 of a locally finite graph , and denote @xmath184 the set of equivalence classes of vertices . 
   such an equivalence relation is called _ edge - compatible _ if whenever @xmath185 , for any vertex @xmath91 the total number of edges from @xmath186 to the members of the equivalence class of @xmath91 equals the total number of edges from @xmath187 to the members of the equivalence class of @xmath91 . 
   when we have such an equivalence relation , we can define a quotient graph @xmath188 with vertex set @xmath184 . 
   namely , we denote for each @xmath189 the respective equivalence classes as @xmath190 $ ] and @xmath191 $ ] , and define the number of edges from @xmath190 $ ] to @xmath191 $ ] in the quotient graph to be @xmath192 \to [ w ] ) : = \sum_{u \in [ w ] } \#(v \to u).\ ] ] by definition of edge - compatibility , the above sum does not depend on the representative @xmath85 chosen inside the class @xmath190 $ ] . 
   moreover , it is easy to see that the quotient map @xmath193 is a weak cover of graphs . 
   let us now relate the growth of a graph to the growth of its weak covers . 
   [ l : growthcover ] let @xmath164 a weak cover of graphs with bounded cycles , and @xmath194 a finite set of vertices of @xmath157 .    1 . 
   suppose that every closed path in @xmath157 passes through @xmath194 . 
   then for each @xmath20 we have the estimate @xmath195 which implies @xmath196 2 . 
   suppose that @xmath197 is a set of closed paths in @xmath158 such that each @xmath198 crosses at least one vertex @xmath91 with the property that : the set @xmath199 is non - empty , and any lift of @xmath24 from an element of @xmath200 ends in @xmath200 . 
   then there exists @xmath201 , which depends on @xmath194 , such that for each @xmath20 we have @xmath202    \(1 ) let @xmath24 be a closed path in @xmath157 of length @xmath20 , based at @xmath85 . 
   let now @xmath203 be the first vertex of @xmath24 which belongs to @xmath194 ( by hypothesis , there is one ) . 
   if we call @xmath204 the cyclical permutation of @xmath24 based at @xmath203 , the projection of @xmath204 to @xmath158 now yields a closed path @xmath205 in @xmath158 which is based at @xmath206 . now note that for each such pair @xmath207 , there are at most @xmath208 possible choices for @xmath209 ; then , given @xmath203 there is a unique lift of @xmath205 from @xmath203 , and @xmath20 possible choices for the vertex @xmath85 on that lift , giving the estimate . 
   \(2 ) recall first that in every directed graph we can define an equivalence relation by saying @xmath210 if there is a path from @xmath85 to @xmath91 and a path from @xmath91 to @xmath85 . 
   the equivalence classes are known as _ strongly connected components _ , or @xmath211 for short . 
   moreover , we can form a graph @xmath212 whose vertices are the strongly connected components and there is an edge from the s.c.c . 
   @xmath213 to @xmath214 if in the original graph there is a path from an element of @xmath213 to an element of @xmath214 . 
   note that by construction this graph has no cycles . 
   moreover , let us consider a pair @xmath215 of distinct elements of @xmath194 ; then either there is no path from @xmath186 to @xmath187 , or we can pick some path from @xmath186 to @xmath187 , which will be denoted @xmath216 . 
   then the set @xmath217 of such paths is finite , and let @xmath72 be the maximum length of an element of @xmath218 . 
   let now @xmath24 be a closed path in @xmath158 based at @xmath219 . by hypothesis 
   , @xmath24 passes through some @xmath91 such that @xmath220 , and each lift of @xmath24 from @xmath200 ends in @xmath200 . 
   let us now consider the set @xmath221 of all s.c.c . of @xmath157 which intersect @xmath200 : 
   since the graph @xmath212 constructed above has no cycles , there is a component @xmath222 which has the property that there is no path from @xmath223 to some other component of @xmath200 . 
   thus , if we pick @xmath224 and lift @xmath24 from @xmath85 to a path @xmath169 in @xmath157 , the endpoint @xmath175 of @xmath169 must lie inside @xmath200 by hypothesis . 
   thus , by the property of @xmath223 , the vertex @xmath175 lies in @xmath223 , hence there is a path @xmath225 from @xmath175 to @xmath85 in the previously chosen set @xmath218 , hence if we take @xmath226 , this is a closed path in @xmath157 of length between @xmath20 and @xmath227 . 
   thus we have a map @xmath228 given by @xmath229 . 
   now , given @xmath85 we can recover @xmath230 , and given @xmath205 we can recover @xmath24 , since @xmath24 is the path given by the first @xmath20 edges of @xmath231 starting at @xmath91 . finally , we have at most @xmath20 choices for the starting point @xmath219 on @xmath24 , hence the fibers of the above map have cardinality at most @xmath20 , proving the claim .    an immediate corollary of the lemma is the following , when @xmath157 and @xmath158 are both finite . 
   [ l : sameentro ] let @xmath156 be finite graphs , and @xmath164 a weak cover . 
   then the growth rate of @xmath157 equals the growth rate of @xmath158 . 
   apply the lemma with @xmath232 , and @xmath197 equal to the set of all closed paths in @xmath158 . 
   let us consider the set @xmath233 of pairs of disjoint positive integers . 
   the set @xmath234 will be sometimes called the _ wedge _ and , given an element @xmath235 , the coordinate @xmath49 will be called the _ height _ of @xmath85 , while the coordinate @xmath110 will be called the _ width _ of @xmath85 . 
   the terminology becomes more clear by looking at figure [ f : wedge ] . 
   @xmath236    we call a _ labeled wedge _ an assignment @xmath237 of a label @xmath238 ( which stands for _ non - separated _ ) or @xmath194 ( which stands for _ separated _ ) to each element of @xmath234 . 
   now , to each labeled wedge @xmath239 we assign a graph @xmath240 in the following way . 
   the vertex set of @xmath240 is the wedge @xmath234 , while the edges of @xmath240 are labeled by the set @xmath241 and determined according to the following rule .    * 
   if @xmath242 is non - separated , then @xmath243 has as its ( unique ) successor the vertex @xmath244 ; we say that the edge @xmath245 is an _ upward edge _ and we label it with @xmath246 ; * if @xmath247 is separated , then @xmath247 has two successors : * * first , we add the edge @xmath248 , which we call _ forward edge _ and label it with @xmath249 . * * second , we add the edge @xmath250 , which we call _ backward edge _ and label it with @xmath251 .    in order to explain the names , note that following an upward or forward edge increases the width by @xmath40 , while following a backward edge ( weakly ) decreases it . moreover , following an upward edge increases the height by @xmath40 , while the targets of both backward and forward edges have height @xmath40 . 
   @xmath252\txt{(2,4 ) }    & ( 2,5 ) & \cdots \\ * + [ f]\txt{(1,2 ) }   & ( 1,3 )    & * + [ f]\txt{(1,4 ) }   & ( 1,5 ) & \cdots \\ } \ ] ]    @xmath253^u & ( 3,5 ) & \cdots \\ & ( 2,3 )   \ar[ur]^u & * + [ f]\txt{(2,4 ) } \ar[dr]^f   \ar@/_/[dl]_b & ( 2,5 ) & \cdots \\ * + [ f]\txt{(1,2 ) } \ar@(u , l)_b   \ar[r]^f & ( 1,3 ) \ar[ur]_u   & * + [ f]\txt{(1,4 ) } \ar[r]^f \ar@/^/[ll]^b & ( 1,5 ) & \cdots \\ } \ ] ]    [ p : properties ] let @xmath240 the graph associated to the labeled wedge @xmath239 
   . then the following are true :    1 . 
   each vertex of a closed path of length @xmath20 has height at most @xmath20 ; 2 . 
   the support of each closed path of length @xmath20 intersects the set @xmath254 ; 3 . 
   each vertex of a closed path of length @xmath20 has width at most @xmath255 ; 4 .   for each @xmath256 
   , there exists at most one separated vertex in the @xmath257 diagonal @xmath258 which is contained in the support of at least one closed path ; 5 .   there are at most @xmath259 multi - cycles of length @xmath20 . 
   note that ( 1 ) , ( 2 ) , ( 3 ) are sharp , as seen by the simple cycle @xmath260^u & 25 \ar@/^/[r]^u   & * + [ f]\txt{36 } \ar@/^/[ll]^b } \ ] ]    \(1 ) let us first note that every closed path contains at least one backward edge , since the upward and forward edges always increase the height . 
   moreover , the endpoint of a backward edge has always height @xmath40 , and each edge increases the height by at most @xmath40 , hence the height of a vertex along the closed path is at most @xmath20 . 
   \(2 ) since the target of each backward edge is @xmath261 , where @xmath49 is the height of the source of the edge , which is at most @xmath20 by the previous point , then the target of each backward edge along the closed path belongs to the set @xmath262 . 
   \(3 ) by the previous point , there is at least a vertex along the closed path with width at most @xmath263 . 
   since every move increases the width by at most @xmath40 , then the largest possible width of a vertex along the path is @xmath264 . 
   \(4 ) let @xmath265 be the separated vertex in @xmath266 with smallest height , if there is one . 
   we claim that no vertex @xmath267 of @xmath266 with @xmath268 belongs to any closed path . note by looking at the rules that , if a vertex @xmath269 of height @xmath270 is the target of some edge , then it must be the target of an upward edge , more precisely an edge from the vertex @xmath271 immediately to the lower left of @xmath85 , which then must be non - separated . 
   thus , since @xmath272 is separated , the vertex @xmath273 does not belong to any closed path ; the claim then follows by induction on @xmath274 , since , by the same reasoning , if @xmath275 belongs to some closed path , then also @xmath276 must belong to the same path . 
   \(5 ) let @xmath277 the backward edges along a multi - cycle @xmath24 of length @xmath20 , and denote @xmath278 the source of @xmath98 , and @xmath279 the height of @xmath278 . 
   note that the set @xmath280 determines @xmath24 , as you can start from @xmath186 , follow the backward edge , and then follow upward or forward edges until you either close the loop or encounter another @xmath278 , and then continue this way until you walk along all of @xmath24 . 
   moreover , we know that for each @xmath278 there are at most @xmath255 possible choices , as each @xmath278 is separated and by ( 4 ) there is at most one for each diagonal @xmath266 , and by ( 3 ) it must lie on some @xmath266 with @xmath281 . 
   we now claim that @xmath282 which is then sufficient to complete the proof . 
   let us now prove the claim . by definition of multi - cycle , then the targets of the @xmath98 must be all distinct , and by the rule these targets are precisely @xmath283 with @xmath284 , hence all @xmath279 must be distinct . moreover , let us note that each @xmath98 must be preceded along the multi - cycle by a sequence @xmath285 of upward edges of length @xmath286 , and all such sequences for distinct @xmath49 must be disjoint . 
   hence we have that all @xmath279 are distinct and their total sum is at most the length of the multi - cycle , i.e. @xmath20 . thus we have @xmath287 which proves the claim . 
   [ t : zero ] let @xmath239 be a labeled wedge . then its associated graph @xmath19 has bounded cycles , and its spectral determinant @xmath21 defines a holomorphic function in the unit disk . 
   moreover , the growth rate @xmath288 of the graph @xmath19 equals the inverse of the smallest real positive root of @xmath129 , in the following sense : @xmath289 for @xmath131 and , if @xmath132 , then @xmath133 .    by construction , 
   the outgoing degree of any vertex of @xmath19 is at most @xmath290 . 
   moreover , by proposition [ p : properties ] ( 5 ) the graph has bounded cycles , and the growth rate of the number @xmath126 of multi - cycles is @xmath291 , since @xmath292{(2n)^{\sqrt{2n } } } = \limsup_{n } e^{\frac{\sqrt{2 } \log(2n)}{\sqrt{n } } } = 1.\ ] ] the claim then follows by theorem [ t : rootofp ] . 
   we shall sometimes denote as @xmath293 the growth rate of the graph associated to the labeled wedge @xmath239 . 
   we say that a sequence @xmath294 of labeled wedges converges to @xmath239 if for each finite set of vertices @xmath295 there exists @xmath238 such for each @xmath296 the labels of the elements of @xmath194 for @xmath297 and @xmath239 are the same . 
   [ l : contwedge ] if a sequence of labeled wedges @xmath294 converges to @xmath239 , then the growth rate of @xmath297 converges to the growth rate of @xmath239 . 
   let @xmath298 and @xmath21 denote respectively the spectral determinants of @xmath297 and @xmath239 . 
   then for each @xmath256 , the coefficient of @xmath299 in @xmath298 converges to the coefficient of @xmath299 in @xmath21 , because by proposition [ p : properties ] ( 1 ) and ( 3 ) the support of any multi - cycle of length @xmath300 is contained in the finite subgraph @xmath301 . 
   thus , since the modulus of the coefficient of @xmath299 is uniformly bounded above by @xmath302 , then @xmath303 uniformly on compact subsets of the unit disk . 
   thus , by rouch s theorem , the smallest real positive zero of @xmath298 converges to the smallest real positive zero of @xmath21 , hence by theorem [ t : zero ] we have @xmath304 . 
   given integers @xmath37 and @xmath38 we define the equivalence relation @xmath305 on @xmath306 by saying that @xmath307 if :    * either @xmath308 and @xmath309 ; * or @xmath310 and @xmath311 
   .    note that if @xmath45 the equivalence relation @xmath312 is simply the congruence modulo @xmath43 . 
   a set of representatives for the equivalence classes of @xmath312 is the set @xmath313 . 
   the equivalence relation induces an equivalence relation on the set @xmath314 of ordered pairs of integers by saying that @xmath315 
   if @xmath316 and @xmath317 . moreover , it also induces an equivalence relation on the set of _ unordered pairs _ of integers by saying that the unordered pair @xmath318 is equivalent to @xmath319 if either @xmath320 or @xmath321 . 
   a labeled wedge is _ periodic _ of period @xmath43 and pre - period @xmath44 if the following two conditions hold :    * any two pairs @xmath247 and @xmath322 such that @xmath323 have the same label ; * if @xmath307 , then the pair @xmath247 is non - separated . 
   if @xmath324 , the labeled wedge will be called _ purely periodic_.    a pair @xmath247 with @xmath325 will be called _ diagonal _ ; hence the second point in the definition can be rephrased as  every diagonal pair is non - separated " . 
   @xmath326 &   \dots \\ & & ( 3,4 )   \ar[ur ]   & ( 3 , 5 ) \ar[ur ]   & \dots\\ & * + [ f]\txt{(2,3 ) } \ar[d ] \ar[dr ] & * + [ f]\txt{(2,4 ) } \ar[dl ] \ar[dr ] & * + [ f]\txt{(2 , 5 ) } \ar[dll ] \ar[dr ] &   \dots \\ ( 1,2 ) \ar[ur ]   & * + [ f]\txt{(1,3 ) } \ar[r ] \ar[l ] & * + [ f]\txt{(1 , 4 ) } \ar[r ] & * + [ f]\txt{(1 , 5 ) } \ar[r ] &   \dots}\ ] ]    @xmath327\txt{(2,3 ) } \ar@<0.5ex>[d ] \ar@<-0.5ex>[d ] \\ ( 1,2 ) \ar[ur ] & * + [ f]\txt{(1,3 ) } \ar[l ]    \ar@(d , r ) } \ ] ]      given a periodic wedge @xmath239 of period @xmath43 and pre - period @xmath44 , with associated ( infinite ) graph @xmath19 , we shall now construct a finite graph @xmath328 which captures the essential features of the infinite graph @xmath19 , in particular its growth rate . 
   the set of vertices of @xmath328 is the set of @xmath312-equivalence classes of non - diagonal , unordered pairs of integers . 
   a set of representatives of @xmath329 is the set @xmath330 the edges of @xmath328 are induced by the edges of @xmath19 , that is are determined by the following rules : if the unordered pair @xmath318 is non - separated , then @xmath318 has one outgoing edge , namely @xmath331 ; while if @xmath318 is separated , then @xmath318 has the two outgoing edges @xmath332 , and @xmath333 .    the main result of this section is the following .    [ 
   p : finiteinfinite ] let @xmath239 be a periodic labeled edge , with associated ( infinite ) graph @xmath19 
   . then the growth rate of @xmath19 equals the growth rate of its finite model @xmath328 .    in order to prove the proposition 
   , we shall also introduce an intermediate finite graph , which we call the _ 
   finite 2-cover _ of @xmath328 , and denote @xmath334 . 
   the set of vertices of @xmath334 is the set of @xmath305-equivalence classes of non - diagonal , ordered pairs of integers , and the edges are induced by the edges of @xmath19 in the usual way . 
   the reason to introduce the intermediate graph @xmath334 is that @xmath328 does not inherit the labeling of edges from @xmath19 , as backward and forward edges in @xmath19 may map to the same edge in @xmath328 , while @xmath334 naturally inherits the labels . 
   let @xmath19 be the graph associated to a periodic labeled wedge .    1 . 
   first , let us observe that no diagonal vertex is contained in the support of any closed path of @xmath19 : in fact , every diagonal vertex is non - separated , and its outgoing edge leads to another diagonal vertex with larger height , hence the path can never close up . 
   thus , we can construct the subgraph @xmath335 by taking as vertices all pairs which are non - diagonal , and as edges all the edges of @xmath19 which do not have either as a source or target a diagonal pair . by 
   what has been just said , the growth rate of @xmath19 and @xmath335 is the same , @xmath336 2 .   since 
   the maps @xmath337 are given by quotienting with respect to equivalence relations , they are both weak covers of graphs . thus , 
   since both @xmath334 and @xmath328 are finite , by lemma [ l : sameentro ] , the growth rates of @xmath334 and @xmath328 are the same . 
   we are now left with proving that the growth rate of @xmath335 is the same as the growth rate of @xmath334 . 
   since the cardinality of the fiber of the projection @xmath338 is infinite , the statement is not immediate . 
   note that the finite @xmath290-cover @xmath334 is a graph with labeled edges : indeed , if @xmath320 , then @xmath339 , and so on , thus the labeling of @xmath334 inherited from @xmath19 is well - defined , and the graph map @xmath338 preserves the labels . 
   4 .   let us call _ backtracking _ a path in @xmath19 or @xmath334 such that at least one of its edges is labeled by @xmath251 (= backward ) , and _ non - backtracking _ otherwise . now let us note the following : 1 . 
   every closed path in @xmath19 is backtracking ; in fact , following any edge which is upward or forward increases the height , thus a path in @xmath19 made entirely of @xmath246 and @xmath249 edges can not close up . 
   2 .   every closed path in @xmath19 passes through the finite set @xmath340 in order to prove this , we first prove that any element in the support of any closed path in @xmath19 has height at most @xmath341 . in order to do so , 
   let us fix a diagonal @xmath342 . 
   by periodicity , either there is a separated pair @xmath247 in @xmath266 of height @xmath343 , or all elements of @xmath266 are non - separated ; in the latter case , no element of @xmath266 is part of any closed path , since any path based at an element of @xmath266 is non - backtracking . in the first case , 
   let @xmath344 be the separated pair with smallest height in @xmath266 ; then , only the elements with height less than @xmath345 can be part of any closed path , and @xmath346 , so the first claim is proven . as a consequence , 
   the target of any @xmath251-labeled edge which belongs to some closed path is of type @xmath347 , where @xmath348 is the height of the source , hence the target belongs to @xmath194 . by the same reasoning , 
   if @xmath24 is any path in @xmath19 based at a vertex of height @xmath40 , then the target of any @xmath251-labeled edge along @xmath24 lies in @xmath194 . 
   3 .   every backtracking closed path in @xmath334 has at least one @xmath251-labeled edge , hence the target of such @xmath251-labeled edge lies in the set @xmath349 , and every lift to @xmath19 starting from an element of @xmath194 must end in the target of a @xmath251-labeled edge in @xmath19 , hence must end in @xmath194 . 
   5 .   finally , let us note that for each @xmath20 , the number of non - backtracking paths of length @xmath20 in @xmath334 is at most the cardinality of @xmath350 ; indeed , from each vertex there is at most one edge labeled @xmath246 or @xmath249 , thus for each vertex of @xmath334 there is at most one non - backtracking path of length @xmath20 based at it . 
   let us now put together the previous statements . 
   indeed , by ( 4)(a)-(b ) every closed path in @xmath19 passes through @xmath194 , hence we can apply lemma [ l : growthcover ] ( 1 ) and get that @xmath351 to prove the other inequality , let us note that by ( 4)(b)-(c ) we know that lemma [ l : growthcover ] ( 2 ) applies with @xmath197 the set of backtracking closed paths in @xmath334 . 
   moreover , by point ( 5 ) above we have that the number of non - backtracking paths is bounded independently of @xmath20 : thus we can write @xmath352 @xmath353 from which follows @xmath354 as required .    when dealing with purely periodic external angles , we shall also need the following lemma . 
   [ l : twoperiodics ] let @xmath355 and @xmath356 be two labeled wedges which are purely periodic of period @xmath43 . 
   suppose moreover that for every pair @xmath247 with @xmath357 the label of @xmath243 in @xmath355 equals the label in @xmath356 
   . then the finite models @xmath358 and @xmath359 are isomorphic graphs . as a consequence , 
   the growth rates of @xmath355 and @xmath356 are equal . 
   let @xmath360 an equivalence class of unordered pairs . 
   if neither @xmath49 nor @xmath110 are divisible by @xmath43 , then the label of @xmath318 is the same in @xmath355 and @xmath356 , hence the outgoing edges from @xmath318 are the same in @xmath358 and @xmath359 . 
   suppose on the other hand that @xmath361 ( hence , @xmath362 because the pair is non - diagonal ) . 
   then , if the pair @xmath363 is non - separated , then its only outgoing edge goes to @xmath364 in @xmath365 . 
   on the other hand , if @xmath363 is separated , then its two possible outgoing edges are @xmath366 and @xmath367 . however , the pair @xmath366 is diagonal , hence no vertex in the graph has such label 
   . thus , independently of whether @xmath363 is separated , it has exactly one outgoing edge with target @xmath367 , proving the claim . 
   we shall now apply the theory of labeled wedges to the core entropy . as we have seen in section [ s : algo ] 
   , thurston s algorithm allows one to compute the core entropy for periodic angles @xmath3 ; in order to interpolate between periodic angles of different periods , we shall now define for _ any _ angle @xmath13 a labeled wedge @xmath27 , and thus an infinite graph @xmath368 as described in the previous sections . 
   recall that for each @xmath31 we denote @xmath50 , which we see as a point in @xmath51 . for each pair @xmath247 which belongs to @xmath234 , we label @xmath247 as _ non - separated _ 
   if @xmath369 and @xmath370 belong to the same element of the partition @xmath46 , or at least one of them lies on the boundary of the partition . 
   if instead @xmath369 and @xmath370 belong to the interiors of two different intervals of @xmath46 , then we label the pair @xmath247 as _ separated_. the labeled wedge just constructed will be denoted @xmath27 , and its associated graph @xmath28 . 
   the main quantity we will work with is the following :              let @xmath3 have period @xmath43 and pre - period @xmath44 . by definition of the equivalence relation @xmath312 , we have @xmath325 if and only if @xmath373 , which proves the first condition in the definition of periodic wedge . moreover , if @xmath247 is a diagonal pair , then @xmath374 , hence the pair @xmath247 is non - separated , verifying the second condition .    using the results of the previous section , we are now ready to prove that the logarithm of the growth rate of @xmath28 coincides with the core entropy for rational angles , proving the first part of theorem [ t : newmain ] . 
   let @xmath3 a rational angle , @xmath28 the infinite graph associated to the labeled wedge @xmath27 , and @xmath376 the finite model of @xmath28 as described in section [ s : finitemodel ] by unraveling the definitions , the matrix @xmath52 constructed in section [ s : algo ] is exactly the adjacency matrix of the finite model @xmath376 . by proposition [ p : finiteinfinite ] , the growth rate of @xmath28 coincides with the growth rate of its finite model @xmath376 . 
   moreover , by lemma [ l : perroneig ] , the growth rate of @xmath376 coincides with the largest real eigenvalue of its adjacency matrix , that is the largest real eigenvalue of @xmath52 . 
   thus , by theorem [ t : algo ] its logarithm is the core entropy @xmath5 .          as @xmath379 , the labeled wedge @xmath380 stabilizes . 
   in fact , for each @xmath49 consider the position of @xmath50 with respect to the partition @xmath46 . if @xmath381 , then , for all @xmath382 in a neighborhood of @xmath3 , @xmath383 lies on the same side of the partition . otherwise , note that for any @xmath384 and @xmath385 , the functions @xmath386 and @xmath387 are both continuous and orientation preserving but have different derivative : thus , for all @xmath388 close enough to @xmath3 , the point @xmath383 lies on one side of the partition , and for @xmath389 close enough to @xmath3 lies on the other side . 
   thus , the limits @xmath390_^+ : = _  ^+ _ , _ ^- : = _  ^- _ @xmath390 exist , and are both equal to @xmath380 if @xmath3 is not purely periodic , because then no @xmath369 with @xmath31 lies on the boundary of the partition . 
   the claim then follows by lemma [ l : contwedge ] . 
   this proves continuity of @xmath371 at all angles which are not purely periodic . 
   we shall now deal with the purely periodic case , where for the moment we only know that the left - hand side and right - hand side limits of @xmath371 exist . 
   [ l : diffpairs ] let @xmath3 be purely periodic of period @xmath43 
   . then @xmath27 , @xmath391 and @xmath392 are purely periodic of period @xmath43 , and differ only in the labelings of pairs @xmath247 with either @xmath361 or @xmath393 . 
   let @xmath3 be purely periodic of period @xmath43 . 
   note that @xmath369 lies on the boundary of the partition @xmath46 if and only if @xmath361 . 
   thus , for all the pairs @xmath247 for which neither component is divisible by @xmath43 , the label of @xmath247 is continuous across @xmath3 , proving the second statement . 
   now we shall show that if for all @xmath388 and close enough to @xmath3 one has @xmath394 on one side of the partition , then also @xmath395 is on the same side for each @xmath256 . 
   this will prove that @xmath396 is purely periodic of period @xmath43 . 
   the proof for @xmath397 is symmetric , and @xmath27 is purely periodic of period @xmath43 by lemma [ l : periodic ] . in order to prove the remaining claim , 
   let us denote @xmath398 , @xmath399 ; since @xmath3 is purely periodic , there exists @xmath400 such that @xmath401 for all @xmath256 . 
   now , note that for each @xmath256 the derivatives satisfy the inequality @xmath402 , thus for each @xmath256 there exists @xmath403 such that for each @xmath404 one has @xmath405 , hence the points @xmath406 all belong to the same side of the partition independently of @xmath300 , as required . 
   [ p : bothsides ] if @xmath3 is purely periodic of period @xmath43 , then the ( infinite ) graphs @xmath28 , @xmath407 and @xmath408 , associated respectively to @xmath27 , @xmath391 and @xmath392 have the same growth rate , i.e. we have the equality @xmath409    by lemma [ l : diffpairs ] and lemma [ l : twoperiodics ] , the graphs @xmath28 , @xmath407 and @xmath408 have the same finite form , hence by proposition [ p : finiteinfinite ] they have the same growth rate , proving the claim . 
   let @xmath2 , and denote by @xmath413 the smallest real zero of @xmath14 . 
   the proof is a simple quantitative version of rouch s theorem . for simplicity , 
   let us assume that @xmath3 is pre - periodic , and let @xmath414 be the minimum distance , in @xmath51 , between ( distinct ) elements of the set @xmath415 . 
   given another angle @xmath416 , denote @xmath417 the smallest real zero of @xmath418 , and choose @xmath20 such that @xmath419 then , by looking at how @xmath369 moves with respect to the partition @xmath46 , we realize that , for @xmath420 , the pair @xmath247 is separated for @xmath3 if and only if it is separated for @xmath416 . by construction , the @xmath421 coefficient of @xmath14 depends only on multi - cycles of total length @xmath20 , and by proposition [ p : properties ] ( 3 ) all such multi - cycles live on the subgraph @xmath422 , hence the first @xmath20 coefficients of @xmath14 and @xmath418 coincide . 
   thus we can write @xmath423 for some @xmath424 and @xmath425 , where we used that @xmath426 by proposition [ p : properties ] ( 5 ) . 
   now , using @xmath427 we have @xmath428 where @xmath256 is the order of zero of @xmath46 at @xmath429 and @xmath430 is a holomorphic function , with @xmath431 . by combining , and , we get @xmath432 with @xmath433 , which implies the claim as @xmath434 . in the periodic case 
   the proof the same , except one should argue separately for @xmath435 and @xmath436 , and use proposition [ p : bothsides ] . 
   a. douady , _ topological entropy of unimodal maps : monotonicity for quadratic polynomials _ , in _ real and complex dynamical systems ( hillerd , 1993 ) _ , nato adv . 
   sci . * 464 * , 6587 .
summary: the core entropy of polynomials , recently introduced by w. thurston , is a dynamical invariant which can be defined purely in combinatorial terms , and provides a useful tool to study parameter spaces of polynomials . 
   the theory of core entropy extends to complex polynomials the entropy theory for real unimodal maps : the real segment is replaced by an invariant tree , known as hubbard tree , which lives inside the filled julia set . 
   we prove that the core entropy of quadratic polynomials varies continuously as a function of the external angle , answering a question of thurston .