id sid tid token lemma pos r494vh57h2d 1 1 the the DET r494vh57h2d 1 2 dynamics dynamic NOUN r494vh57h2d 1 3 of of ADP r494vh57h2d 1 4 a a DET r494vh57h2d 1 5 rational rational ADJ r494vh57h2d 1 6 surface surface NOUN r494vh57h2d 1 7 map map NOUN r494vh57h2d 1 8 φ φ X r494vh57h2d 1 9 : : PUNCT r494vh57h2d 1 10 x x PUNCT r494vh57h2d 1 11 ⇢ ⇢ X r494vh57h2d 1 12 x x PUNCT r494vh57h2d 1 13 are be AUX r494vh57h2d 1 14 easier easy ADJ r494vh57h2d 1 15 to to PART r494vh57h2d 1 16 analyse analyse VERB r494vh57h2d 1 17 when when SCONJ r494vh57h2d 1 18 φ φ PROPN r494vh57h2d 1 19 is be AUX r494vh57h2d 1 20 ' ' PUNCT r494vh57h2d 1 21 algebraically algebraically ADV r494vh57h2d 1 22 stable stable ADJ r494vh57h2d 1 23 ' ' PUNCT r494vh57h2d 1 24 . . PUNCT r494vh57h2d 2 1 in in ADP r494vh57h2d 2 2 this this DET r494vh57h2d 2 3 thesis thesis NOUN r494vh57h2d 2 4 , , PUNCT r494vh57h2d 2 5 we we PRON r494vh57h2d 2 6 discuss discuss VERB r494vh57h2d 2 7 algebraic algebraic ADJ r494vh57h2d 2 8 stability stability NOUN r494vh57h2d 2 9 and and CCONJ r494vh57h2d 2 10 develop develop VERB r494vh57h2d 2 11 tools tool NOUN r494vh57h2d 2 12 from from ADP r494vh57h2d 2 13 non−archimedean non−archimedean ADJ r494vh57h2d 2 14 dynamics dynamic NOUN r494vh57h2d 2 15 in in ADP r494vh57h2d 2 16 order order NOUN r494vh57h2d 2 17 to to PART r494vh57h2d 2 18 understand understand VERB r494vh57h2d 2 19 the the DET r494vh57h2d 2 20 algebraic algebraic ADJ r494vh57h2d 2 21 stability stability NOUN r494vh57h2d 2 22 of of ADP r494vh57h2d 2 23 skew skew ADJ r494vh57h2d 2 24 product product NOUN r494vh57h2d 2 25 maps map NOUN r494vh57h2d 2 26 . . PUNCT r494vh57h2d 3 1 there there PRON r494vh57h2d 3 2 are be VERB r494vh57h2d 3 3 three three NUM r494vh57h2d 3 4 main main ADJ r494vh57h2d 3 5 outcomes outcome NOUN r494vh57h2d 3 6 , , PUNCT r494vh57h2d 3 7 separated separate VERB r494vh57h2d 3 8 into into ADP r494vh57h2d 3 9 four four NUM r494vh57h2d 3 10 chapters chapter NOUN r494vh57h2d 3 11 . . PUNCT r494vh57h2d 4 1 first first ADV r494vh57h2d 4 2 , , PUNCT r494vh57h2d 4 3 we we PRON r494vh57h2d 4 4 investigate investigate VERB r494vh57h2d 4 5 when when SCONJ r494vh57h2d 4 6 and and CCONJ r494vh57h2d 4 7 how how SCONJ r494vh57h2d 4 8 algebraically algebraically ADV r494vh57h2d 4 9 stability stability NOUN r494vh57h2d 4 10 can can AUX r494vh57h2d 4 11 be be AUX r494vh57h2d 4 12 achieved achieve VERB r494vh57h2d 4 13 by by ADP r494vh57h2d 4 14 conjugating conjugate VERB r494vh57h2d 4 15 φ φ PROPN r494vh57h2d 4 16 with with ADP r494vh57h2d 4 17 a a DET r494vh57h2d 4 18 birational birational ADJ r494vh57h2d 4 19 change change NOUN r494vh57h2d 4 20 of of ADP r494vh57h2d 4 21 coordinate coordinate NOUN r494vh57h2d 4 22 . . PUNCT r494vh57h2d 5 1 such such DET r494vh57h2d 5 2 a a DET r494vh57h2d 5 3 conjugacy conjugacy NOUN r494vh57h2d 5 4 is be AUX r494vh57h2d 5 5 called call VERB r494vh57h2d 5 6 an an DET r494vh57h2d 5 7 ' ' PUNCT r494vh57h2d 5 8 * * ADJ r494vh57h2d 5 9 algebraic algebraic ADJ r494vh57h2d 5 10 stabilisation stabilisation NOUN r494vh57h2d 5 11 * * PUNCT r494vh57h2d 5 12 ' ' PUNCT r494vh57h2d 5 13 . . PUNCT r494vh57h2d 6 1 we we PRON r494vh57h2d 6 2 show show VERB r494vh57h2d 6 3 that that SCONJ r494vh57h2d 6 4 if if SCONJ r494vh57h2d 6 5 this this PRON r494vh57h2d 6 6 can can AUX r494vh57h2d 6 7 be be AUX r494vh57h2d 6 8 done do VERB r494vh57h2d 6 9 with with ADP r494vh57h2d 6 10 a a DET r494vh57h2d 6 11 birational birational ADJ r494vh57h2d 6 12 morphism morphism NOUN r494vh57h2d 6 13 , , PUNCT r494vh57h2d 6 14 then then ADV r494vh57h2d 6 15 there there PRON r494vh57h2d 6 16 is be VERB r494vh57h2d 6 17 a a DET r494vh57h2d 6 18 minimal minimal ADJ r494vh57h2d 6 19 such such ADJ r494vh57h2d 6 20 stabilisation stabilisation NOUN r494vh57h2d 6 21 . . PUNCT r494vh57h2d 7 1 for for ADP r494vh57h2d 7 2 birational birational ADJ r494vh57h2d 7 3 φ φ X r494vh57h2d 7 4 ; ; PUNCT r494vh57h2d 7 5 we we PRON r494vh57h2d 7 6 also also ADV r494vh57h2d 7 7 show show VERB r494vh57h2d 7 8 that that SCONJ r494vh57h2d 7 9 repeatedly repeatedly ADV r494vh57h2d 7 10 lifting lift VERB r494vh57h2d 7 11 φ φ PROPN r494vh57h2d 7 12 to to ADP r494vh57h2d 7 13 its its PRON r494vh57h2d 7 14 graph graph NOUN r494vh57h2d 7 15 gives give VERB r494vh57h2d 7 16 an an DET r494vh57h2d 7 17 algebraic algebraic ADJ r494vh57h2d 7 18 stabilisation stabilisation NOUN r494vh57h2d 7 19 . . PUNCT r494vh57h2d 8 1 we we PRON r494vh57h2d 8 2 provide provide VERB r494vh57h2d 8 3 an an DET r494vh57h2d 8 4 example example NOUN r494vh57h2d 8 5 in in ADP r494vh57h2d 8 6 which which PRON r494vh57h2d 8 7 φ φ X r494vh57h2d 8 8 can can AUX r494vh57h2d 8 9 be be AUX r494vh57h2d 8 10 birationally birationally ADV r494vh57h2d 8 11 conjugated conjugate VERB r494vh57h2d 8 12 to to ADP r494vh57h2d 8 13 a a DET r494vh57h2d 8 14 stable stable ADJ r494vh57h2d 8 15 map map NOUN r494vh57h2d 8 16 , , PUNCT r494vh57h2d 8 17 but but CCONJ r494vh57h2d 8 18 the the DET r494vh57h2d 8 19 conjugacy conjugacy NOUN r494vh57h2d 8 20 can can AUX r494vh57h2d 8 21 not not PART r494vh57h2d 8 22 be be AUX r494vh57h2d 8 23 achieved achieve VERB r494vh57h2d 8 24 solely solely ADV r494vh57h2d 8 25 by by ADP r494vh57h2d 8 26 blowing blow VERB r494vh57h2d 8 27 up up ADP r494vh57h2d 8 28 . . PUNCT r494vh57h2d 9 1 second second ADV r494vh57h2d 9 2 , , PUNCT r494vh57h2d 9 3 we we PRON r494vh57h2d 9 4 develop develop VERB r494vh57h2d 9 5 a a DET r494vh57h2d 9 6 dynamical dynamical ADJ r494vh57h2d 9 7 theory theory NOUN r494vh57h2d 9 8 for for ADP r494vh57h2d 9 9 what what PRON r494vh57h2d 9 10 we we PRON r494vh57h2d 9 11 call call VERB r494vh57h2d 9 12 a a DET r494vh57h2d 9 13 ' ' PUNCT r494vh57h2d 9 14 skew skew ADJ r494vh57h2d 9 15 product product NOUN r494vh57h2d 9 16 ' ' PUNCT r494vh57h2d 9 17 on on ADP r494vh57h2d 9 18 the the DET r494vh57h2d 9 19 berkovich berkovich PROPN r494vh57h2d 9 20 projective projective ADJ r494vh57h2d 9 21 line line NOUN r494vh57h2d 9 22 , , PUNCT r494vh57h2d 9 23 φ\_\ φ\_\ PROPN r494vh57h2d 9 24 * * NOUN r494vh57h2d 9 25 : : PUNCT r494vh57h2d 9 26 ℙ¹(k ℙ¹(k SPACE r494vh57h2d 9 27 ) ) PUNCT r494vh57h2d 9 28 ⇢ ⇢ PROPN r494vh57h2d 9 29 ℙ¹(k ℙ¹(k SPACE r494vh57h2d 9 30 ) ) PUNCT r494vh57h2d 9 31 . . PUNCT r494vh57h2d 10 1 skew skew ADJ r494vh57h2d 10 2 products product NOUN r494vh57h2d 10 3 strictly strictly ADV r494vh57h2d 10 4 generalise generalise VERB r494vh57h2d 10 5 the the DET r494vh57h2d 10 6 notion notion NOUN r494vh57h2d 10 7 of of ADP r494vh57h2d 10 8 a a DET r494vh57h2d 10 9 ' ' PUNCT r494vh57h2d 10 10 * * ADJ r494vh57h2d 10 11 rational rational ADJ r494vh57h2d 10 12 map map NOUN r494vh57h2d 10 13 * * PUNCT r494vh57h2d 10 14 ' ' PUNCT r494vh57h2d 10 15 on on ADP r494vh57h2d 10 16 ℙ¹(k ℙ¹(k SPACE r494vh57h2d 10 17 ) ) PUNCT r494vh57h2d 10 18 . . PUNCT r494vh57h2d 11 1 we we PRON r494vh57h2d 11 2 describe describe VERB r494vh57h2d 11 3 the the DET r494vh57h2d 11 4 analytical analytical ADJ r494vh57h2d 11 5 , , PUNCT r494vh57h2d 11 6 algebraic algebraic ADJ r494vh57h2d 11 7 , , PUNCT r494vh57h2d 11 8 and and CCONJ r494vh57h2d 11 9 dynamical dynamical ADJ r494vh57h2d 11 10 properties property NOUN r494vh57h2d 11 11 of of ADP r494vh57h2d 11 12 skew skew ADJ r494vh57h2d 11 13 products product NOUN r494vh57h2d 11 14 , , PUNCT r494vh57h2d 11 15 including include VERB r494vh57h2d 11 16 a a DET r494vh57h2d 11 17 study study NOUN r494vh57h2d 11 18 of of ADP r494vh57h2d 11 19 periodic periodic ADJ r494vh57h2d 11 20 points point NOUN r494vh57h2d 11 21 , , PUNCT r494vh57h2d 11 22 and and CCONJ r494vh57h2d 11 23 a a DET r494vh57h2d 11 24 fatou fatou X r494vh57h2d 11 25 / / SYM r494vh57h2d 11 26 julia julia PROPN r494vh57h2d 11 27 dichotomy dichotomy PROPN r494vh57h2d 11 28 . . PUNCT r494vh57h2d 12 1 the the DET r494vh57h2d 12 2 chapter chapter NOUN r494vh57h2d 12 3 ends end VERB r494vh57h2d 12 4 with with ADP r494vh57h2d 12 5 the the DET r494vh57h2d 12 6 classification classification NOUN r494vh57h2d 12 7 of of ADP r494vh57h2d 12 8 the the DET r494vh57h2d 12 9 connected connect VERB r494vh57h2d 12 10 components component NOUN r494vh57h2d 12 11 of of ADP r494vh57h2d 12 12 the the DET r494vh57h2d 12 13 fatou fatou PROPN r494vh57h2d 12 14 set set PROPN r494vh57h2d 12 15 . . PUNCT r494vh57h2d 13 1 third third ADV r494vh57h2d 13 2 we we PRON r494vh57h2d 13 3 consider consider VERB r494vh57h2d 13 4 algebraic algebraic ADJ r494vh57h2d 13 5 stabilisation stabilisation NOUN r494vh57h2d 13 6 for for ADP r494vh57h2d 13 7 a a DET r494vh57h2d 13 8 skew skew ADJ r494vh57h2d 13 9 product product NOUN r494vh57h2d 13 10 on on ADP r494vh57h2d 13 11 a a DET r494vh57h2d 13 12 ruled rule VERB r494vh57h2d 13 13 surface surface NOUN r494vh57h2d 13 14 φ φ X r494vh57h2d 13 15 : : PUNCT r494vh57h2d 13 16 x x PUNCT r494vh57h2d 13 17 ⇢ ⇢ X r494vh57h2d 13 18 x x PUNCT r494vh57h2d 13 19 classically classically ADV r494vh57h2d 13 20 this this PRON r494vh57h2d 13 21 is be AUX r494vh57h2d 13 22 a a DET r494vh57h2d 13 23 mapping mapping NOUN r494vh57h2d 13 24 of of ADP r494vh57h2d 13 25 the the DET r494vh57h2d 13 26 form form NOUN r494vh57h2d 13 27 φ(x φ(x PROPN r494vh57h2d 13 28 , , PUNCT r494vh57h2d 13 29 y y PROPN r494vh57h2d 13 30 ) ) PUNCT r494vh57h2d 13 31 = = PROPN r494vh57h2d 14 1 ( ( PUNCT r494vh57h2d 14 2 φ1(x φ1(x SPACE r494vh57h2d 14 3 ) ) PUNCT r494vh57h2d 14 4 , , PUNCT r494vh57h2d 14 5 φ2(x φ2(x SPACE r494vh57h2d 14 6 , , PUNCT r494vh57h2d 14 7 y y PROPN r494vh57h2d 14 8 ) ) PUNCT r494vh57h2d 14 9 ) ) PUNCT r494vh57h2d 14 10 . . PUNCT r494vh57h2d 15 1 the the DET r494vh57h2d 15 2 main main ADJ r494vh57h2d 15 3 result result NOUN r494vh57h2d 15 4 is be AUX r494vh57h2d 15 5 that that SCONJ r494vh57h2d 15 6 when when SCONJ r494vh57h2d 15 7 φ1 φ1 PROPN r494vh57h2d 15 8 has have VERB r494vh57h2d 15 9 no no DET r494vh57h2d 15 10 superattracting superattracte VERB r494vh57h2d 15 11 cycles cycle NOUN r494vh57h2d 15 12 , , PUNCT r494vh57h2d 15 13 then then ADV r494vh57h2d 15 14 we we PRON r494vh57h2d 15 15 can can AUX r494vh57h2d 15 16 always always ADV r494vh57h2d 15 17 find find VERB r494vh57h2d 15 18 an an DET r494vh57h2d 15 19 algebraic algebraic ADJ r494vh57h2d 15 20 stabilisation stabilisation NOUN r494vh57h2d 15 21 . . PUNCT r494vh57h2d 16 1 we we PRON r494vh57h2d 16 2 provide provide VERB r494vh57h2d 16 3 an an DET r494vh57h2d 16 4 example example NOUN r494vh57h2d 16 5 of of ADP r494vh57h2d 16 6 a a DET r494vh57h2d 16 7 skew skew ADJ r494vh57h2d 16 8 product product NOUN r494vh57h2d 16 9 φ φ X r494vh57h2d 16 10 where where SCONJ r494vh57h2d 16 11 φ1 φ1 PROPN r494vh57h2d 16 12 has have VERB r494vh57h2d 16 13 a a DET r494vh57h2d 16 14 superattracting superattracte VERB r494vh57h2d 16 15 fixed fix VERB r494vh57h2d 16 16 point point NOUN r494vh57h2d 16 17 and and CCONJ r494vh57h2d 16 18 φ φ NOUN r494vh57h2d 16 19 is be AUX r494vh57h2d 16 20 not not PART r494vh57h2d 16 21 algebraically algebraically ADV r494vh57h2d 16 22 stable stable ADJ r494vh57h2d 16 23 in in ADP r494vh57h2d 16 24 any any DET r494vh57h2d 16 25 model model NOUN r494vh57h2d 16 26 . . PUNCT r494vh57h2d 17 1 any any DET r494vh57h2d 17 2 product product NOUN r494vh57h2d 17 3 φ φ X r494vh57h2d 17 4 : : PUNCT r494vh57h2d 17 5 x x PUNCT r494vh57h2d 17 6 ⇢ ⇢ X r494vh57h2d 17 7 x x PUNCT r494vh57h2d 17 8 of of ADP r494vh57h2d 17 9 this this DET r494vh57h2d 17 10 kind kind NOUN r494vh57h2d 17 11 induces induce VERB r494vh57h2d 17 12 one one NUM r494vh57h2d 17 13 φ\_\ φ\_\ PROPN r494vh57h2d 17 14 * * NOUN r494vh57h2d 17 15 : : PUNCT r494vh57h2d 17 16 ℙ¹(𝕂 ℙ¹(𝕂 NUM r494vh57h2d 17 17 ) ) PUNCT r494vh57h2d 17 18 ⇢ ⇢ PROPN r494vh57h2d 17 19 ℙ¹(𝕂 ℙ¹(𝕂 NUM r494vh57h2d 17 20 ) ) PUNCT r494vh57h2d 17 21 on on ADP r494vh57h2d 17 22 the the DET r494vh57h2d 17 23 berkovich berkovich PROPN r494vh57h2d 17 24 projective projective ADJ r494vh57h2d 17 25 line line NOUN r494vh57h2d 17 26 over over ADP r494vh57h2d 17 27 the the DET r494vh57h2d 17 28 field field NOUN r494vh57h2d 17 29 𝕂 𝕂 PROPN r494vh57h2d 17 30 of of ADP r494vh57h2d 17 31 puiseux puiseux PROPN r494vh57h2d 17 32 series series PROPN r494vh57h2d 17 33 ; ; PUNCT r494vh57h2d 17 34 this this DET r494vh57h2d 17 35 φ\_\ φ\_\ NOUN r494vh57h2d 17 36 * * NOUN r494vh57h2d 17 37 describes describe VERB r494vh57h2d 17 38 φ φ PROPN r494vh57h2d 17 39 on on ADP r494vh57h2d 17 40 the the DET r494vh57h2d 17 41 completion completion NOUN r494vh57h2d 17 42 of of ADP r494vh57h2d 17 43 a a DET r494vh57h2d 17 44 fibre fibre NOUN r494vh57h2d 17 45 of of ADP r494vh57h2d 17 46 xb xb PROPN r494vh57h2d 17 47 . . PUNCT r494vh57h2d 18 1 the the DET r494vh57h2d 18 2 fatou fatou PROPN r494vh57h2d 18 3 / / SYM r494vh57h2d 18 4 julia julia PROPN r494vh57h2d 18 5 theory theory NOUN r494vh57h2d 18 6 for for ADP r494vh57h2d 18 7 φ\_\ φ\_\ PROPN r494vh57h2d 18 8 * * NOUN r494vh57h2d 18 9 is be AUX r494vh57h2d 18 10 instrumental instrumental ADJ r494vh57h2d 18 11 to to PART r494vh57h2d 18 12 understand understand VERB r494vh57h2d 18 13 algebraic algebraic ADJ r494vh57h2d 18 14 stabilisation stabilisation NOUN r494vh57h2d 18 15 for for ADP r494vh57h2d 18 16 φ φ PROPN r494vh57h2d 18 17 . . PROPN r494vh57h2d 19 1 one one NUM r494vh57h2d 19 2 chapter chapter NOUN r494vh57h2d 19 3 is be AUX r494vh57h2d 19 4 devoted devoted ADJ r494vh57h2d 19 5 to to ADP r494vh57h2d 19 6 specialising specialise VERB r494vh57h2d 19 7 these these DET r494vh57h2d 19 8 non−archimedean non−archimedean ADJ r494vh57h2d 19 9 tools tool NOUN r494vh57h2d 19 10 to to ADP r494vh57h2d 19 11 the the DET r494vh57h2d 19 12 case case NOUN r494vh57h2d 19 13 where where SCONJ r494vh57h2d 19 14 k k PROPN r494vh57h2d 19 15 = = PROPN r494vh57h2d 19 16 𝕂 𝕂 PROPN r494vh57h2d 19 17 and and CCONJ r494vh57h2d 19 18 describing describe VERB r494vh57h2d 19 19 the the DET r494vh57h2d 19 20 transfer transfer NOUN r494vh57h2d 19 21 of of ADP r494vh57h2d 19 22 geometric geometric ADJ r494vh57h2d 19 23 information information NOUN r494vh57h2d 19 24 from from ADP r494vh57h2d 19 25 one one NUM r494vh57h2d 19 26 world world NOUN r494vh57h2d 19 27 and and CCONJ r494vh57h2d 19 28 skew skew ADJ r494vh57h2d 19 29 product product NOUN r494vh57h2d 19 30 to to ADP r494vh57h2d 19 31 another another PRON r494vh57h2d 19 32 . . PUNCT r494vh57h2d 20 1 we we PRON r494vh57h2d 20 2 include include VERB r494vh57h2d 20 3 a a DET r494vh57h2d 20 4 description description NOUN r494vh57h2d 20 5 of of ADP r494vh57h2d 20 6 free free ADJ r494vh57h2d 20 7 and and CCONJ r494vh57h2d 20 8 satellite satellite NOUN r494vh57h2d 20 9 blowups blowup NOUN r494vh57h2d 20 10 in in ADP r494vh57h2d 20 11 the the DET r494vh57h2d 20 12 dual dual ADJ r494vh57h2d 20 13 graph graph NOUN r494vh57h2d 20 14 of of ADP r494vh57h2d 20 15 divisors divisor NOUN r494vh57h2d 20 16 on on ADP r494vh57h2d 20 17 xb xb PROPN r494vh57h2d 20 18 . . PUNCT