............................................................................

"We have not succeeded in answering all our problems. The answers we have found only serve to raise a whole set of new questions. In some ways we feel we are as confused as ever, but we believe we are confused on a higher level and about more important things."

Posted outside the mathematics reading room, Tromso University

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............................................................................

Curriculum Vitae de Artur O. Lopes ( pdf file )

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Sugestao de livros para formar uma biblioteca basica de Matematica por Artur O. Lopes (htm file)

.......................................................................

The list of books by Artur Lopes and related texts by other authors

.......................................................................

** papers (pdf files) **

joint work with Rafael Ruggiero

preprint (2022)

joint work with Jader E. Brasil and Josue Knorst

preprint (2022)

119) "Dynamical hypothesis tests and Decision Theory for Gibbs distributions",

joint work with M. Denker and S. R. C. Lopes

preprint (2021)

joint work with M. Karling and S. R. C. Lopes

preprint (2021)

joint work with L. Cioletti, L. Hataishi and M. Stadlbauert

preprint (2018)

joint work with R. Ruggiero,

preprint (2018)

joint work Jairo Mengue

Disc. and Cont. Dyn. Syst. Series A (on line)

joint work with Jader Brasil and Josue Knorst,

Communications in Contemporary Mathematics (on line)

113) "Entropy, pressure, ground states and calibrated sub-actions for linear dynamics",

joint work with V. Vargas

Bull. of the Braz. Math. Soc. Vol (on line)

joint work with H. H. Ferreira and S. R. C. Lopes

Journal of Dynamics and Games, Volume 9, Number 2, April 2022 - pp 125-150

MR4408114

111) "Bayes posterior convergence for loss functions via almost additive Thermodynamic Formalism"

joint work S. R. C. Lopes and P. Varandas

Journ. of Statis. Physics, 186:35 (2022)

MR4375840

110) "Pentadiagonal Matrices and an Application to the Centered MA(1) Stationary Gaussian Process",

joint work with M. Karling and S. R. C. Lopes

International Journal of Applied Mathematics and Statistics, Vol. 61; Issue No. 1, pp 1-22 (2022)

109) "Nonequilibrium in Thermodynamic Formalism: the Second Law, gases and Information Geometry"

joint work with R. Ruggiero

Qualitative Theory of Dynamical Systems 21: 21 p 1-44 (2022) MR4356488

108) "Lyapunov exponents for Quantum Channels: an entropy formula and generic properties"

joint work Jader E. Brasil, and Josue Knorst

Journal of Dynamical Systems and Geometric Theories, Vol 19(2) 155-187 (2021) MR4358481

107) "Invariant probabilities for discrete time Linear Dynamics via Thermodynamic Formalism"

joint work A. Messaoudi, M. Stadlbauert and V. Vargas

Nonlinearity 34 (2021) 8359–-8391 MR4344719

106) "An iterative process for approximating subactions"

joint work with Hermes H. Ferreira and Elismar R. Oliveira,

Modeling, Dynamics, Optimization and Bioeconomics IV" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, Springer Verlag, pp 187--212 (2021)

MR4378497

105) "On Bertelson-Gromov Dynamical Morse Entropy ",

joint work with Marcos Sebastiani,

"Modeling, Dynamics, Optimization and Bioeconomics IV" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, Springer Verlag, 297--321, (2021)

MR4378502

joint work with G. Castro and G. Mantovani

"Modeling, Dynamics, Optimization and Bioeconomics IV" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, Springer Verlag 79--137 (2021)

MR4378494

joint work with Jairo Mengue,

Erg. Theo. and Dyn. Systems, (2021), 41, 1835–1863 MR4252210

102) "Quantum Spin probabilities at positive temperature are Holder Gibbs probabilities"

joint work with J. Brasil, J. Mengue and Carlos G. Moreira

Comm. in Contemporary Mathematics Vol. 23, No. 1 (2021) 1950050 (32 pages) MR4169241

101) "Explicit examples in Ergodic Optimization"

joint work with Hermes H. Ferreira and Elismar R. Oliveira,

Sao Paulo Journal of Math. Sciences - Vol 14 - pp 443-489 (2020) MR4173468

100) "The Ruelle operator for symmetric beta-shifts"

joint work with Victor Vargas,

Publicacions Mathematiques, 661 - 680 - Volume 64, Issue 2 (2020) MR4119264

99) "Gibbs States and Gibbsian Specifications on the space R^N"

joint work with Victor Vargas,

Dyn. Systems., Volume 35, Issue 2, 216-241 (2020) MR4120828

98) "Ruelle Operator for Continuous Potentials and DLR-Gibbs Measures",

joint work with L. Cioletti and M. Stadlbauert

Disc and .Cont. Dyn. Syst. A Vol 40, N. 8, 4625-4652 (2020) MR4112024

97) "Correlation Inequalities and Monotonicity Properties of the Ruelle Operator",

joint work with L. Cioletti

Stoch. and Dyn, 19 (2019), no. 6, 1950048, 31 pp MR4033007

joint work with Elismar Oliveira

Bull of the Braz. Math. Soc., Vol 50. Issue 3, pp 663–683 (2019) MR3993188

95) "The KMS Condition for the homoclinic equivalence relation and Gibbs probabilities",

joint work with G. Mantovani

Sao Paulo Journ. of Math. Sciences, Volume 13, Issue 1, pp 248–282 (2019) MR3947403

94) "A general renormalization procedure on the one-dimensional lattice and decay of correlations",

Stoch. and Dyn., Vol. 19, No. 01, 1950003 (2019) MR3904031

93) "The calculus of thermodynamical formalism",

joint work with B. Kloeckner, P. Giulietti and D. Marcon,

Journ. of the European Math Society, Vol 20, Issue 10, pages 2357-2412 (2018) MR3852182

92) "Large Deviations for Quantum Spin probabilities at temperature zero",

joint work with Jairo K. Mengue, Joana Mohr and Carlos G. Moreira,

Stochastics and Dynamics, Vol. 18, No. 06, 1850044 (2018) MR3869882

91) "Selection of calibrated subaction when temperature goes to zero in the discounted problem",

joint work with Renato Iturriaga and Jairo Mengue

Discrete and Cont Dyn. Syst., Series A, Vol 38, n 10, 4997-5010 (2018) MR3834704

90) "A formula for the Entropy of the Convolution of Gibbs probabilities on the circle",

Nonlinearity, 31 , 3441-3459 (2018) MR3816762

89) "Uma breve Introdução à Matemática da Mecânica Quântica"

- XXXI Coloquio Brasileiro de Matematica - 2017 MR3753695

88) "Introdução à Matemática da Mecânica Quântica"

On line text

87) "On the thin boundary of the fat attractor"

joint work with E. Oliveira,

"Modeling, Dynamics, Optimization and Bioeconomics III" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, Springer Verlag, 205-246 (2018) MR3806667

86) "Transport and large deviations for Schrodinger operators and Mather measures",

joint work with Ph. Thieullen, "Modeling, Dynamics, Optimization and Bioeconomics III" Editors: Alberto Pinto and David Zilberman, Proceedings in Mathematics and Statistics, Springer Verlag, 247-255 (2018) MR3806668

85) "An introduction to Coupling"

"Modeling, Dynamics, Optimization and Bioeconomics II" Editors: Alberto Pinto and David Zilberman, Springer Proceedings in Mathematics and Statistics, pp 307-335, (2017) Springer Verlag MR3718407

84) "A detailed proof of the von Neumann's Quantum Ergodic Theorem",

joint work with Marcos Sebastiani,

Quantum Studies: Mathematics and Foundations - Vol 4 - Issue 3, 263-285 (2017) MR3692490

joint work with L. Cioletti, Discrete and Cont. Dyn. Syst. - Series A, Vol 37, Number 12, 6139 -- 6152 (2017) MR3690296

82) "Spectral Properties of the Ruelle Operator for Product Type Potentials on Shift Spaces",

joint work with L. Cioletti, M. Denker and M. Stadlbauer, Journal of the London Mathematical Society - Volume 95, Issue 2, 684–704 (2017) MR3656287

81) "Semiclassical limits, Lagrangian states and coboundary equations",

joint work with J. Mohr,

Vol 17. N 2, 1750014 (19 pages) Stoch. and Dynamics (2017) MR3626745

joint work with P. Giulietti and V. Pit

Stoch. and Dynamics, 16 , 1660011 - 22 -pages - vol 3 (2016) MR3471364

joint work with Benoit Kloeckner and Manuel Stadlbauert,

Nonlinearity, 28, Number 11, 4117–4137 (2015) MR3424906

78) "Generic properties for random repeated quantum iterations",

joint work with Marcos Sebastiani,

Quantum Studies: Mathematics and Foundations, Vol 2, pp 389-402 (2015) MR3412580

77) "Entropy, Pressure and Duality for Gibbs plans in Ergodic Transport"

joint work with J. K. Mengue, J. Mohr and R. R. Souza - Bull. of the Brazilian Math. Soc. Vol 46 - N 3 - 353-389 (2015) MR3403546

joint work with J. K. Mengue, J. Mohr and R. R. Souza

Erg. Theory and Dyn Systems, 35 (6), 1925–1961 (2015) MR3377291

75) "The Dual Potential, the involution kernel and Transport in Ergodic Optimization"

joint work with E. Oliveira and Ph. Thieullen,

Dynamics, Games and Science -International Conference and Advanced School Planet Earth DGS II, Portugal (2013), Edit. J-P Bourguignon, R. Jelstch, A. Pinto and M. Viana, Springer Verlag, pp 357-398 (2015) MR3675753

74) "Phase Transitions in One-dimensional Translation Invariant Systems: a Ruelle Operator Approach"

joint work with L. Cioletti,

Journ. of Statistical Physics, 159 - Issue 6, 1424–1455 (2015) MR3350377

joint work with A. Neumann,

Journ. of Statistical Physics. Vol. 159 - Issue 4 pp 797-822 (2015) MR3336981

joint work with Elismar R. Oliveira

Dynamical Systems: an Int. Journ. Vol 30. Issue 1, 1-12 (2015) MR3304972

71) "Ergodic Transport Theory, periodic maximizing probabilities and the twist condition"

joint work with G. Contreras and E. Oliveira,

"Modeling, Optimization, Dynamics and Bioeconomy I", Springer Proceedings in Mathematics and Statistics, Volume 73, Edit. David Zilberman and Alberto Pinto, 183-219 (2014) MR3297616

70) "Selection of measure and a Large Deviation Principle for the general one-dimensional XY model"

joint work with J. Mengue,

Dyn. Systems: an Int. Jour. Vol 29, Issue 1 (2014) pp 24-39 MR3170635

69) "On the selection of subaction and measure for a subclass of potentials defined by P. Walters"

joint work with A. T. Baraviera and J. Mengue,

Erg. Theo. and Dyn. Systems, Volume 33, issue 05, pp. 1338--1362 (2013) MR3103086

68) "Ergodic Optimization, Zero Temperature Limits and the Max-Plus Algebra"

joint work with A. Baraviera and R. Leplaideur, mini-course in XXIX Coloquio Brasileiro de Matemática (2013) MR3114331

joint work with A. Neumann and Ph. Thieullen, Journ. of Statist. Phys. Volume 152, Issue 5, Page 894-933 (2013) MR3101487

66) "The effective potential and transshipment in thermodynamic formalism at temperature zero"

joint work with E. Garibaldi,

Stoch. and Dynamics, Vol 13 - N 1, 1250009 (13 pages) (2013) MR3007247

65) "Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics"

joint work with E. R. Oliveira and D. Smania,

Bull. of the Braz. Math Soc. Vol 43 (3) 467-512 (2012) MR3024067

64) "Duality Theorems in Ergodic Transport"

joint work with J. Mengue, Journal of Statistical Physics. Vol 149, issue 5, pp 921–942 (2012) MR2999567

63) "Wigner measures and the semi-classical limit to the Aubry-Mather measure"

joint work with Diogo Gomes and Joana Mohr,

Applied Mathematics Research Express, Volume 2012, Issue 2, 152-183 (2012) MR2982772

62) "The potential point of view for renormalization",

joint work with A Baraviera and R. Leplaideur

Stochastic and Dynamics, Vol 12. N 4, (2012) 1250005(1-34) MR2990585

joint work with A. Baraviera and R. Leplaideur,

SIAM Journal on Applied Dynamical Systems, Vol. 11, n 1, 243-260 (2012) MR2902616

joint work with A. T. Baraviera, L. M. Cioletti, J. Mohr, R. R. Souza,

Reviews in Math. Physics. Vol. 23, N. 10, pp 1063–-1113 (2011) MR2864625

59) "A dynamical point of view of Quantum Information: Wigner measures",

joint work with A. Baraviera, C. F. Lardizabal, and M. Terra Cunha.

Dynamics, Games and Science II, DYNA 2008, Edit. M. Peixoto, A. Pinto and D Rand, pp 161-186 Springer Verlag (2011) MR2883277

58) "A dynamical point of view of Quantum Information: entropy and pressure",

joint work with A. Baraviera, C. F. Lardizabal, and M. Terra Cunha.

Dynamics, Games and Science I, DYNA 2008, Edit. M. Peixoto, A. Pinto and D Rand, pp 81-122 Springer Verlag (2011) MR3051429

57) "Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics"

joint work with R. Ruggiero,

Disc. and Cont. Dyn. Syst. A. Vol 29, N 3, pp 1155-1174 -(2011) MR2773169

56) "The Mather measure and a Large Deviation Principle for the Entropy Penalized Method",

joint work with D. Gomes and J. Mohr

"Communications in Contemporary Mathematics", Vol 13, issue 2, 235-268 (2011) MR2794485

55) "Quantum Stochastic Processes, Quantum Iterated Function Systems and Entropy"

joint work with A. Baraviera, C. F. Lardizabal and M. Terra Cunha, Sao Paulo Journal of Math. Sciences, Vol. 5, N 1, pp 51-84 (2011) MR2859245

54) "Zeta measures and Thermodynamic Formalism for temperature zero"

joint work with J. Mengue

Bull. of the Bras. Math. Soc 41 (3) pp 449-480 (2010) MR2718151

53) "A Thermodynamic Formalism for density matrices in Quantum Information",

joint work with A. Baraviera, C. F. Lardizabal and M. Terra Cunha.

Applied Mathematics Research Express, Vol. 2010, No. 1, pp. 63–118 (2010) MR2629875

52) "Poncelet pairs and the Twist Map associated to the Poncelet Billiard",

joint work with M. Sebastiani

"Real Analysis Exchange", Vol 35 , Issue 2, pp 355-374 (2010) MR2683603

51) "A Ruelle Operator for continuous time Markov Chains"

joint work with A. Baraviera and R. Exel

Sao Paulo Journal of Mathematical Sciences, vol 4 n. 1, pp 1-16 (2010) MR2760963

50) "Minimax Probabilities for Aubry-Mather problems",

joint work with Diogo Gomes and N. Jung

, Commun. Contemporary Mathematics, Vol 12, Issue 5, pp 789-813 MR2733198 (2010)

49) "On calibrated and separating sub-actions",

joint work with E. Garibaldi and P. Thieullen

Bull. of the Bras. Math. Soc. Vol 40, 577-602, (4)(2009) MR2563132

48) "Entropy and variational principles for holonomic probabilities of IFS",

joint work with E. Oliveira

"Disc. and Cont. Dyn. Systems" (Series A) Vol 23, N 3, 937-955 (2009) MR2461833

47) "KMS States, Entropy and a Variational Principle for Pressure",

joint work with G. Castro

"Real Analysis Exchange" Vol 34 , Issue 2, 333-346 (2009) MR2569191

46) " Negative Entropy, Zero temperature and stationary Markov chains on the interval",

joint work with J. Mohr, R. Souza and Ph. Thieullen

Bull. Soc. Bras. Math. Vol 40 n 1, (2009), 1-52 MR2496111

45) "C*- Algebras and Thermodynamic Formalism",

joint work with R. Exel

Sao Paulo Journal of Mathematical Sciences 2, 1 (2008), 285–307 MR2503825

joint work with Diogo Gomes

Sao Paulo Journal of Mathematical Sciences 2, 1 (2008), 21-28 MR2502621

43) "Eigenfunctions of the Laplacian and associated Ruelle operator ",

joint work with P. Thieullen

, "Nonlinearity", Volume 21, Number 10, pp-2239-2254 (2008) MR2439478

42) "On the Aubry-Mather theory for symbolic dynamics",

joint work with E. Garibaldi

Ergodic Theory and Dynamical Systems, v. 28, issue 3, pp 791-815 (2008) MR2422016

41) "Functions for relative maximization",

joint work with E. Garibaldi

Dynamical Systems, v. 22, 511-528, 2007 MR2377213

40) "Cohomology and Subcohomology for expansive geodesic flows",

joint work with V. Rosas e R. Ruggiero

"Discrete and Continous Dynamical Systems" Volume: 17, N. 2, pp 403-422 (2007) MR2257442

joint work with A. Baraviera e P. Thieullen

"Stochastics and Dynamics" Vol 6, n.1,- pp 77-96 (2006) MR2210682

the Bowen-Series transformation,",

joint work with P. Thieullen

"Annal Inst Henry Poincare - Anal non-linear". pp 663-682, vol 5, (2006) MR2259611

trabalho conjunto com P. Thieullen

Erg Theo and Dyn Syst vol 25 Issue 2, pp605-628 (2005) MR2129112

36) ``C^* Algebras, approximately proper equivalence

relations and Thermodynamic Formalism",

trabalho conjunto com R. Exel

Vol 24, pp 1051-1082 Erg Theo and Dyn Syst (2004) MR2085390

trabalho conjunto com P. Thieullen

Asterisque Vol 287 pp 135-146 ( 2003) MR2040005

34) ``Convergence in distribution

of the periodogram of chaotic systems,"

trabalho conjunto com S. Lopes

"Stochastic and Dynamics", vol 2, Issue 4 pp 609-624 (2002) MR1949304

33) ``Exact bounds for the polynomial

decay of correlation, 1/f noise and the CLT for a non-Holder potential,",

trabalho conjunto com A. Fisher

"Nonlinearity", vol 14, pp 1071-1104 (2001) MR1862813

32) ``Lyapunov minimizing measures

for expanding maps of the circle",

trabalho conjunto com G. Contreras and P. Thieullen

"Ergodic Theory and Dynamical Systems" , vol 21, Issue 5 pp 1379-1409 (2001) MR1855838

31) `` The capacity costfunction of a hard constrained channel"

trabalho conjunto com M. Craizer

publicado em Int Journal of Appl Math Vol 2, N 10 pp 1165-1180 (2000) MR1775717

30) ``Large deviations for expanding transformations with additive white noise"

trabalho conjunto com S. Carmona, publicado em

J. Statist. Phys. 98 (2000), no. 5-6, 1311–-1333 MR1751702

29)"On the minimal action function of an autonomous Lagrangian associated with a magnetic phenomena"

trabalho conjunto com Mario Carneiro, "Annal Inst Henry Poincare - Anal Non-lin" Vol 16, N 6 , pp 667-690 (1999) MR1720511

28) ``A level-1 LDP for the autocovariances

of uniquely ergodic transformations with aditive noise,",

joint work with S. Carmona, C. Landim and S. Lopes

"Jour. of Statist. Physics", Vol. 91, N. 1/2, pp. 395-421 (1998) MR1632522

27) ``Parametric estimation and spectral analysis of piecewise linear maps of the interval. "

joint work with Silvia Lopes,

Adv. in Appl. Probab. 30 (1998), no. 3, 757–776 MR1663557

26)" Generic Properties of open billiards"

trabalho conjunto com M. Mauldin, "Journal of Mathematical Analysis and Applications", Vol 205 pp 251-258 (1997) MR1426992

25) "A billiard in the hyperbolic plane with decay of correlation of type n^-2"

trabalho conjunto M. Bauer, "Discrete and Continuous Dynamical Systems", v. 3, n.1, p. 107-116, 1997 MR1422542

24) "On the spectral density of a class of chaotic time series",

trabalho conjunto com S.Lopes e R. Souza , "Journal of Times Series", Vol 18 N 5, spt pp 465-474 (1997) MR1482452

23) "Spectral Analysis of one-dimensional chaotic transformations" "

trabalho conjunto com S. Lopes e R. Souza, "Random and Computational Dynamics Journal" Vol 3, N 1 pp 107-117 (1997) MR1429841

22) ``Open biliards: invariant and conditionally invariant probabilities on

trabalho conjunto com R. Markarian,

"SIAM Journal of Applied Mathematics", Vol 56, Issue 2, april, pp 651-680, 1996 MR1381665

21) ``On the dimension of the predictive process of a memoryless channel",

trabalho conjunto com Silvia Lopes , "Stochastic Analysis and Applications",

Vol 14, N 4 (1996) MR1402687

20)) `` Spectral analysis of chaotic transformations.",

trabalho conjunto com S. Lopes and R. Souza,

Rebrape 10 (1996), no. 2, 151–179 MR1429841

19) ``The Theta Group and the Continued

Fraction Expansion with Even Partial Quotients",

trabalho conjunto com C. Kraaikamp

Geometricae Dedicatae , vol 59, pp 293-333 1996 MR1371228

18) ``Maximum Likelihood and Minimum Entropy

trabalho conjunto com P. Collet

e A. Galves. Random and Computational Dynamics, 3, pp-241-256, 1995 MR1362772

17)``Absolutely Continuous Invariant Measures for a Class of Affine Interval Exchange Maps",

trabalho conjunto com Z. Coelho e L. F. da Rocha, " Proceedings of the American Mathematical Society", 123, N. 11, pp. 3533-3542, 1995 MR1322918

16) ``Invariant Measure for the Gauss Map Associated with Interval Exchange Maps",

trabalho conjunto com L. F. da Rocha, Indiana University Mathematics Journal, Vol. 13, N. 4, pp. 1399-1438, (1994) MR1322625

16) ``Entropy, Pressure and Large Deviation'',

in Cellular Automata, Dynamical Systems and Neural Networks, E. Goles e S. Martinez (eds.), Kluwer, pp. 79--146, (1994) MR1304502

15) "Weight-balanced measures and free energy for one-dimensional dynamics",

joint work with W. Withers, Forum Math., Vol. 5, pp. 161-182 (1993) MR1205251

14) "The Zeta Function, Non-Differentiability of Pressure and The Critical Exponent of Transition",

"Advances in Mathematics", Vol. 101, pp. 133-167, (1993) MR1242602

13) `On the Dynamics of Real Polynomials on the Plane",

Computers and Graphics, Vol. 16, N. 1, pp. 15-23, (1992).

12) ``A First-Order Level-2 Phase Transition in Thermodynamic Formalism",

Journal of Statistical Physics, Vol. 60, N. 3/4, pp. 395-411, (1990) MR1069638

11) ``Dynamics of Real Polynomials on the Plane and Triple Point Phase Transition",

Mathematical and Computer Modelling, Vol. 13, N. 9, pp. 17-32, (1990) MR1091924

10)"Entropy and Large Deviation",

"NonLinearity", Vol. 3, N 2, pp. 527-546, (1990) MR1054587

9) "Dimension Spectra and a Mathematical Model for Phase Transitions",

Adv. in Appl. Math. 11, pp 475-502 (1990) MR1077258

8) ``A note on delta equilibrium for rational maps",

Mathematische Zeitschrift, 202, 261-269 (1989) MR1013088

7) `The Dimension Spectrum of the Maximal Measure",

SIAM Journal of Mathematical Analysis, Vol. 20, N. 5, pp. 1243-1254, (1989) MR1009356

6)"An Analogy of The Charge Distribution on Julia Sets with the Brownian Motion",

"Journal of Mathematical Physics", Vol. 30, N 9, pp. 2120-2124, (1989) MR1009928

5)" The Complex Potential generated by the maximal measure for a family of rational maps",

"Journal of Statistical Physics", Vol. 52, N 3/4, pp. 571-575, (1988) MR0968947

4) `Equilibrium Measures for Rational Maps",

Ergodic Theory and Dynamical Systems, Vol. 6, Part 3, pp. 393-399, (1986) MR0863202

3) "Orthogonality and the Hausdorff Dimension of the Maximal Measure",

" Proceedings of the American Mathematical Society", Vol. 98, N 1, 51-55, (1986) MR0848874

2) `An Invariant Measure for Rational Maps",

Boletim da Sociedade Brasileira de Matem\'atica, Vol. 14, N. 1, pp. 45-62, Alexandre Freire, Artur O. Lopes and e Ricardo Mane (1983) MR0736568

1)`Structural Stability and Hyperbolic Attractors",

Transactions of the American Mathematical Society, Vol. 252, pp. 205-219, (1979) MR0534118