Volume 4

CRM MONOGRAPH SERIES Centre d e Recherches Mathematique s Universite d e Montrea l

Dynamical Zet a Function s for Piecewis e Monoton e Maps o f th e Interva l

David Ruell e

The Centr e d e Recherche s Mathematique s (CRM ) o f th e Universite d e Montrea l wa s create d i n 196 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l programs , an d publishing. Th e CR M i s supporte d b y th e Universit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , an d th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h th e Institu t de s Science s Mathematiques (ISM ) o f Montreal , whos e constituen t members ar e Concordi a University , McGil l University , th e Universite d e Montreal , th e Universit e d u Quebe c a Montreal, an d th e Ecol e Polytechnique .

^ H E M ^

American Mathematical Societ y Providence, Rhode Island US A

^NDED

https://doi.org/10.1090/crmm/004

The productio n o f thi s volum e was supported i n par t b y th e Chair e Andr e Aisenstadt , the Fond s pou r l a Formatio n d e Chercheur s e t PAid e a l a Recherch e (Fond s FCAR ) an d the Natura l Science s an d Engineerin g Researc h Counci l o f Canad a (NSERC) .

2000 Mathematics Subject Classification. Primar y 58-XX .

Library o f Congres s Cataloging-in-Publicatio n Dat a Ruelle, David .

Dynamical zet a function s fo r piecewis e monotone map s of the interva l / Davi d Ruelle . p. cm. — (CRM monograp h series ; ISSN 1065-8599 ; v. 4)

Originally published : Providence , RI : American Mathematica l Society , ©1994 . Includes bibliographica l references . ISBN 0-8218-3601-3 (softcover ) 1. Differentiabl e dynamica l systems . 2 . Functions , Zeta . 3 . Mapping s (Mathematics )

4. Monotone operators . I . Title . II . Series.

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10 9 8 7 6 5 4 3 2 1 0 9 08 07 06 05 04

Contents

Foreword

Chaptei §1. §2. §3-§4. §5-§6.

§7. §8. §9.

§10. §11-§12. §13. §14. §15.

Chaptei

§1. §2-§3. §4-§5. §6-§7.

: 1 . A n Introductio n t o Dynamica l Zet a Function s Counting periodi c orbit s fo r map s an d flows. Subshifts o f finite type . The produc t formul a fo r maps . The produc t formul a fo r semiflows . The Lefschet z formula. Historical note : Fro m th e Rieman n zet a functio n t o dynamical zet a functions . Properties o f dynamical zet a functions . Transfer operators . Traces an d determinants . Entire analyti c functions . The theor y o f Fredholm-Grothendieck . Analyticity improvin g linea r maps . Non-Fredholm situations . Thermodynamic formalism . Ties with othe r part s o f mathematics .

: 2 . Piec e wise Monotone Map s Definitions. Construction o f new systems . The functiona l 0 . The transfe r operato r £ . Zeta functions . Thermodynamic formalism . Appendix: Extensio n o f the definitio n o f pressure .

Bibliography

V

1 1 2 3 4 5

7 9

10 11 12 13 16 18 19 21

23 23 25 34 38 44 53 58

61

111

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Foreword

The presen t monograp h i s base d o n th e Aisenstad t lecture s give n by th e autho r i n Octobe r 199 3 at th e Universit e d e Montrea l o n "Dy -namical Zet a Functions" . Bu t th e emphasi s i s different . O n on e han d two excellen t review s o f th e subjec t alread y exist , du e t o Parr y an d Pollicott [33] , and t o Balad i [3] . O n the othe r han d th e theory o f zet a functions fo r hyperboli c dynamica l systems is in a state o f flux becaus e of curren t wor k b y Rug h [45 ] an d Pried . Hyperboli c system s ar e thu s not discusse d i n detai l here . Afte r a genera l introductio n (Chapte r 1 ) we concentrat e o n piecewis e monoton e map s o f the interval , an d giv e a detaile d proo f o f a generalize d for m o f th e theore m o f Balad i an d Keller [4 ] (Chapte r 2) . Th e Baladi-Kelle r theore m i s typica l o f wha t one wants t o prov e abou t zet a functions associate d wit h variou s kind s of dynamica l systems , an d th e versio n presente d her e appear s reason -ably final .

October 199 3 Davi d Ruell e

V

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Bibliography

1. M . Arti n an d B . Mazur , On periodic points, Ann . o f Math . (2 ) 8 1 (1965) , 82-99 . 2. M . Atiya h an d R . Bott , A Lefschetz fixed point formula for elliptic complexes, Ann . o f Math .

86 (1967) , 374-407 ; 8 8 (1968) , 451-491 . 3. V . Baladi , Dynamical zeta functions, Rea l an d Comple x Dynamica l System s (B . Branner an d

P. Hjorth , eds.) , Kluwe r Academi c Publisher s (t o b e published) . 4. V . Balad i an d G . Keller , Zeta functions and transfer operators for piecewise monotone trans-

formations, Comm . Math . Phys . 12 7 (1990) , 459-477 . 5. V . Balad i an d D . Ruelle , An extension of the theorem of Milnor and Thurston on zeta func-

tions of interval maps, Ergodi c Theor y Dynamica l System s (t o appear) . 6. , Some properties of zeta functions associated with maps in one dimension (i n prepa -

ration). 7. P . Billingsley , Ergodic Theory and Information, Joh n Wiley , Ne w York , 1965 . 8. R . Bowen , Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lectur e

Notes i n Math . vol . 470 , Springer-Verlag , Berlin , 1975 . 9. R . Bowe n an d O.E . Lanford , Zeta functions of restrictions of the shift transformation, Globa l

Analysis, Proc . Symp . Pur e Math . vol . 14 , Amer. Math . Soc , Providence , R.I . (1975) , pp . 43 -49.

10. G . Choque t an d P.-A . Meyer , Existence et unicite des representations integrates dans les convexes compacts quelconques, Ann . Inst . Fourie r (Grenoble ) 1 3 (1963) , 139-154 .

11. M . Denker , C . Grillenberge r an d K . Sigmund , Ergodic theory on compact spaces, Lectur e Notes i n Math . vol . 527 , Springer-Verlag , Berlin , 1976 .

12. D . Fried, The zeta functions of Ruelle and SelbergI, Ann . Sci . Ecole Norm. Sup . (4 ) 1 9 (1986) , 491-517.

13. , Rationality for isolated expansive sets, Adv . i n Math . 6 5 (1987) , 35-38 . 14. , The flat-trace asymptotics of a uniform system of contractions (Preprint) . 15. A . Grothendieck , Produits tensoriels topologiques et espaces nucleaires, Mem . Amer . Math .

Soc. vol . 16 , Providence , R.I , 1955 . 16. , La theorie de Fredholm, Bull . Soc . Math . Franc e 8 4 (1956) , 319-384 . 17. J . Guckenheimer , Axiom A-\- no cycles = ^ C/ W rational, Bull . Amer . Math . Soc . 7 6 (1970) ,

592-594. 18. V . Guillemi n an d Sh . Sternberg , Geometric asymptotics, Math . Survey s vol . 14 , Amer . Math .

Soc , Providence , R.I. , 1977 . 19. N . Haydn , Meromorphic extension of the zeta function for Axiom A flows, Ergodi c Theor y

Dynamical System s 1 0 (1990) , 347-360 . 20. F . Hofbauer , Piecewise invertible dynamical systems, Probab . Theor . Relat . Field s 7 2 (1986) ,

359-386. 21. F . Hofbaue r an d G . Keller , Zeta-functions and transfer-operators for piecewise linear trans-

formations, J . Rein e Angew . Math . 35 2 (1984) , 100-113 . 22. G . Kelle r an d T . Nowicki , Spectral theory, zeta functions and the distribution of periodic

points for Collet-Eckmann maps, Comm . Math . Phys . 14 9 (1992) , 31-69 . 23. G . Levin , M . Sodi n an d P . Yuditskii , A Ruelle operator for a real Julia set, Comm . Math .

Phys. 14 1 (1991) , 119-131 . 24. , Ruelle operators with rational weights for Julia sets, J . Analys e Math , (t o appear) . 25. A . Manning , Axiom A diffeomorphisms have rational zeta functions, Bull . Londo n Math . Soc .

3 (1971) , 215-220 . 26. M . Martens , Interval dynamics, Thesis , Delft , 1990 .

61

62 B I B L I O G R A P H Y

27. D . Mayer , Continued fractions and related transformations, Ergodi c Theory , Symboli c Dy -namics an d Hyperboli c Space s (T . Bedford , M . Keane , C . Series , eds. ) Oxfor d Universit y Press, Oxford , 1991 .

28. W . d e Melo , Lectures on one-dimensional dynamics, 17 e Coloqui o Brasileir o d e Matematica , Rio d e Janeiro .

29. J . Milno r an d W . Thurston , On iterated maps of the interval, Dynamica l Systems , Lectur e Notes i n Mathematic s vol . 1342 , Springer , Berlin , 1988 , pp . 465-563 .

30. Niho n Sugakkai , ed. , Encyclopedic Dictionary of Mathematics, MI T Press , Cambridge , Mass. , 1977.

31. R.D . Nussbaum , The radius of the essential spectrum, Duk e Math . J . 3 7 (1970) , 473-478 . 32. W . Parr y an d M . Pollicott , An analogue of the prime number theorem for closed orbits of

Axiom A flows, Ann . o f Math . (2 ) 11 8 (1983) , 573-591 . 33. , Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Societ e

Mathematique d e Franc e (Asterisqu e vol . 187-188) , Paris , 1990 . 34. C.J . Preston , Iterates of maps on an interval, Lectur e Notes in Mathematics vol . 999, Springer ,

Berlin, 1983 . 35. D . Ruelle , Statistical mechanics on a compact set with Z v action satisfying expansiveness and

specification, Bull . Amer . Math . Soc . 7 8 (1972) , 988-991 ; Trans . AM S 18 5 (1973) , 237-251 . 36. , Zeta functions and statistical mechanics, Asterisqu e 4 0 (1976) , 167-176 . 37. , Generalized zeta-functions for axiom A basic sets, Bull . Amer . Math . Soc . 82 (1976) ,

153-156. 38. , Zeta-functions for expanding maps and Anosov flows, Invent . Math . 3 4 (1976) , 231-

242. 39. , Thermodynamic Formalism, Addison-Wesley , Readin g MA , 1978 . 40. , The thermodynamic formalism for expanding maps, Comm . Math . Phys . 12 5 (1989) ,

239-262. 41. , An extension of the theory of Fredholm determinants, Inst . Haute s Etude s Sci . Publ .

Math. 7 2 (1990) , 175-193 . 42. , Spectral properties of a class of operators associated with maps in one dimension,

Ergodic Theor y Dynamica l System s 1 1 (1991) , 757-767 . 43. , Analytic completion for dynamical zeta functions, Helv . Phys . Act a 6 6 (1993) , 181 -

191. 44. , Functional equation for dynamical zeta functions of Milnor-Thurston type (t o ap -

pear). 45. H.H . Rugh , The correlation spectrum, for hyperbolic analytic maps, Nonlinearit y 5 (1992) ,

1237-1263. 46. S . Smale , Differentiate dynamical systems, Bull . Amer . Math . Soc . 7 3 (1967) , 747-817 . 47. F . Tangerman , Meromorphic continuation of Ruelle zeta functions, Bosto n Universit y thesis ,

1986 (unpublished) . 48. P . Walters , Ergodic Theory. Introductory Lectures, Lectur e Note s i n Math . vol . 458, Springer -

Verlag, Berlin , 1975 . 49. , A variational principle for the pressure of continuous transformations, Amer . J .

Math. 9 7 (1976) , 937-971 .

Titles in This Series Volume 4 Davi d Ruelle

Dynamical zeta functions fo r piecewise monotone map s of the interva l 1994

3 V . Kumar Murty Introduction to Abelian varietie s 1993

2 M . Ya. Antimirov, A. A. Kolyshkin, and Remi Vaillancourt Applied integral transform s 1993

1 D . V. Voiculescu, K. J. Dykema, and A. Nica Free random variables 1992