Most of the papers since 1990, in their prelimenary form, can be also downloaded from
the IMS Preprint server or from the archive.

Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I--II

General overview of Low-Dimensional Dynamics:

Analytic low-dimensional dynamics: From dimension one to two.
Proceedings of the ICM-14, v.1, 443--474.
Based on the plenary talk available on YouTube

Recent releases.

Renormalization of unicritical diffeomorphisms of the disk.
arXiv: 2401.13559 (2024) (joint with S. Crovissier, E. Pujals and J. Yang).

Structure of hyperbolic polynomial automorphisms of C2 with disconnected Julia set.>
arXiv: 2309.14135 (2023) (joint with R. Dujardain).

Schwarz reflections and the Tricorn.
arXiv:1812.01573 (2018);
to appear in Ann. Inst. Fourier (2024)
(joint with S.-Y. Lee, N. Makarov, and S. Mukherjee).

David extension of circle homeomorphisms: welding, mating, and removability.
arXiv:2010.11256 (2020);
to appear in Memoirs AMS (2024)
(joint with S. Merenkov, S. Mukherjee, and D. Ntalampekos).

Dynamics of Schwarz reflections: the mating phenomena.
arXiv:1811.04979 (2018);
Ann. Sci. Ec. Norm. Sup., v.56 (2023), 1825--1881
(joint with S.-Y. Lee, N. Makarov, and S. Mukherjee ).

Antiholomorphic correspondences and mating I: realization theorems.
(2023) (joint with J. Mazor and S. Mukherjee) .

Uniform a priori bounds for neutral renormalization.
(2023) (joint with Dzmitry Dudko ).

Local connectivity of the Mandelbrot set at some satellite parameters of bounded type.
GAFA online (2023) (joint with Dzmitry Dudko ).

On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections.
Conformal Geometry and Dynamics v.27 (2023), 1--54 (joint with R. Lodge, S. Merenkov, and S. Mukherjee ).

Self-similar groups and holomorphic dynamics: renormalization, integrability, and spectrum.
Arnold Math. J. online (2023) (joint with N.-B. Dang and R. Grigorchuk ).

Lebesgue measure of Feigenbaum Julia sets.
Annals. Math., v 195 (2022), 1--88 (joint with Artur Avila ).

Structure of partially hyperbolic H\'enon maps.
JEMS, v. 23 (2021), 3075--3128 (joint with Han Peters ).

Schwarz reflections and anti-holomorphic correspondences.
Advances Math., v. 385 (2021) (joint with S.-Y. Lee, N. Makarov, and S. Mukherjee).

The critical locus of H\'enon maps.
arXiv:2101.12148 (2021) (joint with John Robertson).

Pacman renormalization and self-similarity of the Mandelbrot set near Siegel parameters.
JAMS, v. 33 (2020), 653--733 (joint with Dzmitry Dudko and Nikita Selinger).

Lee-Yang zeros for the Diamond Hierarchical Lattice and 2D rational dynamics, II: Global pluripotential interpretation.
J. Geometric Analysis, v. 30 (2020), 777-833 (joint with P. Bleher and R. Roeder).

Hedgehogs for neutral dissipative germs of holomorphic diffeomorphisms of (C2,0)
In: ``Quelques aspects de la th\'eorie des syst\`ems dynamiques: un hommage \`a Jean-Christoph Yoccoz (vol II)''.
Asterisque, v. 416 (2020), 193--211 (joint with Tanya Firsova, Remus Radu and Raluca Tanase ).

Hedgehogs in higher dimensions and their applications
In: ``Quelques aspects de la th\'eorie des syst\`ems dynamiques: un hommage \`a Jean-Christoph Yoccoz (vol II)''.
Asterisque, v. 416 (2020), 213--251 (joint with Remus Radu and Raluca Tanase ).

Schwarz reflections, rational maps, Kleinian groups and correspondences.

[1] Dynamics of Schwarz reflections: the mating phenomena.
arXiv:1811.04979 (2018);
Ann. Sci. Ec. Norm. Sup., v.56 (2023), 1825--1881
(joint with S.-Y. Lee, N. Makarov, and S. Mukherjee ).

[2] Schwarz reflections and the Tricorn.
arXiv:1812.01573 (2018);
to appear in Ann. Inst. Fourier (2024)
(joint with S.-Y. Lee, N. Makarov, and S. Mukherjee).

[3] On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections.
Conformal Geometry and Dynamics v.27 (2023), 1--54 (joint with R. Lodge, S. Merenkov, and S. Mukherjee ).

[4] David extension of circle homeomorphisms: welding, mating, and removability.
arXiv:2010.11256 (2020);
to appear in Memoirs AMS (2024)
(joint with S. Merenkov, S. Mukherjee, and D. Ntalampekos).

[5] Antiholomorphic correspondences and mating I: realization theorems.
(2023) (joint with J. Mazor and S. Mukherjee) .

[6]

Real quadratic family.

[1] The Fibonacci unimodal map.
J. Amer. Math. Soc., v. 6 (1993), # 2, 425-457 (joint with John Milnor ).

[2] Combinatorics, geometry and attractors of quasi-quadratic maps.
Annals of Mathematics, v. 140 (1994), 347-404.
See also: Note on the geometry of generalized parabolic towers.

[3] Dynamics of quadratic polynomials, I-II.
Acta Mathematica, v. 178 (1997), 185-297.

[4] Dynamics of quadratic polynomials: Complex bounds for real maps.
Annalles de l'Institut Fourier, v. 47 (1997), # 4, 1219 - 1255 (joint with Michael Yampolsky ).

[6] Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture.
Annals of Mathematics, v. 149 (1999), 319 - 420.

[5] Dynamics of quadratic polynomials III: parapuzzle and SBR measures.
Asterisque volume in honor of Adrien Douady's 60th birthday
``G\'eom\'etrie complexe et syst\'emes dynamiques'', v. 261 (2000), 173 - 200.

[7] Almost every real quadratic map is either regular or stochastic.
Annals of Mathematics, v. 156 (2002), 1 - 78.

[8] The quadratic family as a qualitatively solvable model of chaos.
Notices of the American Math. Society, October 2000.

Comments
More general unicritical dynamics
• Teichmuller space of Fibonacci maps.
  Preprint IMS at Stony Brook, 1993, # 12.

• Regular or stochastic dynamics in real analytic families of unimodal maps.
  Inventiones Math., v. 154 (2003), 451 -- 550 (joint with A. Avila and W. de Melo ).

• Parapuzzle of the Multibrot set and typical dynamics of unimodal maps.
  Journal of European Math Soc., v. 13 (2011), 27--56. (joint with A. Avila and W. Shen ).

• The full renormalization horseshoe for unimodal maps of higher degree: Exponential contraction along hybrid classes.
  Publications IHES, 114 (2012), p. 171--223 (joint with Artur Avila).

• Fourty years of unimodal dynamics: On the occasion Artur Avila winning the Brin prize.
  Journal of Modern Dynamics., v. 6 (2012), 183--203.
  
  Steps towards MLC

  All papers in this series are joint with Jeremy Kahn.

• Quasi-Additivity Law in conformal geometry.
  Annals Math., v. 169 (2009), 561--593.

• Local connectivity of Julia sets for unictritical polynomials.
  Annals of Math., v. 170 (2009), 413--426.

• Combinatorial rigidity for unicritical polynomials.
  Annals of Math., v. 170 (2009), 783--797 (joint with A. Avila and W. Shen).

• A priori bounds for some infinitely renormalizable quadratics, II. Decorations.
  Ann. Sci. Ecole Norm. Sup., v. 41 (2008), 57--84.

• A priori bounds for some infinitely renormalizable quadratics, III. Molecules.
  In: ``Complex Dynamics: Families and Friends''.
  Proceeding of the conference dedicated to Hubbard's 60th birthday
  (ed.: D. Schleicher ). Peters, A K, Limited, 2009.
  
  Complex Henon family

• Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents.
  Inventions Math., v. 112 (1993), 77 - 125 (joint with Eric Bedford and John Smillie).

• Distribution of periodic points of polynomial diffeomorphisms of C^2.
  Inventions Math., v. 114 (1993), 277 - 288 (joint with E. Bedford and J. Smillie) .

• Stability and bifurcations for dissipative polynomial automodphisms of C^2.
  Inventiones Math., v. 200 (2015), 439--511 (joint with Romain Dujardin).

• Classification of invariant Fatou components for dissipative Henon maps.
  GAFA, v. 24 (2014), 887--915 (joint with Han Peters).

  "lambda-lemma for families of Riemann surfaces and the critical loci of complex H\'enon map"
  Preprint IMS at Stony Brook, #3 (2014) (joint with Tanya Firsova ).
  

  Real Henon family.

• Renormalization in the Henon family, I. Universality but non-rigidity.
  J. Stat.Phys., Special issue dedicated to Feigenbaum's 60th birthday,
  v. 121 (2005), 611--669 (joint with A. de Carvalho and M. Martens).

• Renormalization in the Henon family, II: the heteroclinic web.
  Inventiones Math., v. 186 (2011), 115--189. (joint with Marco Martens).

• Probabilistic universality on two-dimensional dynamics.
  Preprint IMS at Stony Brook, # 2 (2011) (joint with M. Martens).
  
  Ising model and rational dynamics

• The Julia sets and complex singularities in hierarchical Ising models,
  Commun. Math. Phys. v. 141 (1992), 453--474. (joint with Pavel Bleher)

• Lee-Yang zeros for DHL and 2D rational dynamics, I. Foliation of the physical cylinder.
  Preprint IMS at Stony Brook 2010 # 4 (joint with P. Bleher and R. Roeder )

• Lee-Yang zeros for DHL and 2D rational dynamics, II. Global pluripotential interpretation.
  Preprint IMS at Stony Brook 2011 # 3 (joint with P. Bleher and R. Roeder )
  
  Measure and dimension of some Julia sets and parameter sets

  [1] Typical behaviour of trajectories of a rational mapping of the sphere.
  Dokl. Akad. Nauk SSSR, v. 268 (1982), 29 - 32.

  [2] On the Lebesgue measure of the Julia set of a quadratic polynomial.
  Preprint IMS at Stony Brook, 1991, # 10.

  [3] How big is the set of infinitely renormalizable quadratics? The volume
  "Voronezh Winter Mathematical Schools" in honor of 80th birthday of S.G. Krein.
  AMS Transl. (2), v. 184 (1998), 131 - 143.

  [4] Hausdorff dimension and conformal measures of Feigenbaum Julia sets.
  J. of the AMS, 21 (2008), 305--383 (joint with A. Avila ).

  [5] Lebesgue measure of Feigenbaum Julia sets.
  arXiv: 1504.02986 [mathDS] (2015) (joint with Artur Avila ).

  Comments
  Laminations
  Laminations in holomorphic dynamics.
  J. Diff. Geom., v. 47 (1997), 17 - 94 (joint with Yair Minsky).

  Conformal and harmonic measures on laminations associated with rational maps.
  Memoirs of the AMS, v. 173 (2005), No 820 (joint with Vadim Kaimanovich).

  Laminations and Holomorphic Dynamics.
  Lecture Notes of the mini-course given at the Conference
  ``New Directions in Dynamical Systems" in Kyoto, August 2002.

  A note on hyperbolic leaves and wild laminations of rational functions.
  Journal of Differential Equations, a volume in honor of Bob Devaney 60th birthday,
  v. 16 (2010), 655--665 (joint with J. Kahn and L. Rempe).
  

  Smooth Ergodic Theory on the interval

  All papers in this section, except [7], are joint with Sasha Blokh.

  [1] Attractors of transformations of the interval.
  Functional Analysis and Appl., v. 21 (1987), 70 - 71.

  [2] Ergodicity of transitive unimodal transformations of the interval.
  Ukrainian Math. J., v. 41 (1989), No 7, 985 - 988

  [3] Attractors of maps of the interval. Banach Center Publ., v. 23 (1989), 427 - 442.

  [4] On the decomposition of one-dimensional dynamical systems into ergodic components.
  Algebra and Analysis, v.1 (1989), 128 - 145.
  English translation: Leningrad Math. J., v. 1 (1989), 137 - 155.

  [4] Measure of solenoidal attractors of unimodal transformations of the interval.
  Math. Notes., v. 48 (1990), No 5, 15 - 20.

  [5] Measure and dimension of solenoidal attractors of one-dimensional dynamical systems.
  Comm. Math. Phys., v. 127 (1990), 573-583.

  [6] Measurable dynamics of S-unimodal maps of the interval.
  Annalles Scientifique Ecole Normale Sup., v. 24 (1991), 545-573.

  [7] Ergodic Theory for smooth one-dimensional dynamical systems.
  Preprint IMS at Stony Brook, 1991, No 11.

  Comments

  No Wandering Intervals

  Non-existence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems,
  I. The case of negative Schwarzian derivative.
  Ergodic Theory and Dynamical Systems, v. 9 (1989), No 4, 737-750.

  Non-existence of wandering intervals and structure of topological attractors of one-dimensional dynamical systems,
  II. The smooth case.
  Ergodic Theory and Dynamical Systems, v. 9 (1989), No 4, 751 - 758 (joint with A. Blokh).

  Transcendental Dynamics

  [1] Iterations of entire functions.
  Dokl. Akad. Nauk SSSR, v. 279 (1984), No 1, 25-27 (joint with Alex Eremenko).

  [2] Examples of entire functions with pathological dynamics.
  J. London Math. Soc., v. 36 (1987), 458 - 468 (joint with Alex Eremenko).

  [3] Dynamical properties of some classes of entire functions.
  Ann. Inst. Fourier, v. 42 (1992), No 4, 989-1020 (joint with Alex Eremenko).

  [4] On typical behaviour of trajectories of the exponential function.
  Russian Math, Surveys, v. 41 (1986), 199 - 200.

  [5] Measurable dynamics of the exponential.
  Siberian Math. J., v. 28 (1987), No 5, 111 - 127.

  [6] Repelling periodic points and landing of rays for post-singularly bounded exponential maps
  Ann. Inst Fourier, v. 64 (2014), 1493--1520 (joint with Anna Benini).

  Comments
  

  Structural stability of rational endomorphisms

  [1] Some typical properties of the dynamics of rational maps.
  Russian Math. Surveys, v. 38 (1983), No 5, 154-155.

  [2] An analysis of stability of the dynamics of rational functions.
  Teoriya Funk., Funk. Anal. and Prilozh., 42 (1984), 72 - 91 (Russian).
  English translation: Selecta Mathematica Sovetica, v. 9 (1990), 69 - 90.

  Comments
  

  Entropy properties of rational endomorphisms

  [1] Entropy of analytic endomorphisms of the Riemann sphere.
  Functional Analysis & Appl., v. 15 (1981), No 4, 83-84.

  [2] The measure of maximal entropy of a rational endomorphism of the Riemann sphere.
  Functional Analysis and Appl., v. 16 (1982), No 4, 78 - 79.

  [3] Entropy properties of rational endomorphisms of the Riemann sphere.
  Ergodic Theory & Dynamical Systems, v. 3 (1983), No 3, 351-385.

  Comments
  

  Various

  
  On the logarithmic property of the degree of a finite group.
  Dokl. Akad. Nauk SSSR, v. 247 (1979), 791--794.

  On cycles and coverings associated to a knot.
  arXiv math 1 301.2205v1 (2013) (joint with Lilya Lyubich).

  Quasisymmetries of Sierpinski carpet Julia sets
  Preprint IMS at Stony Brook, #1 (2014) (joint with Mario Bonk and Sergei Merenkov> )
  

  Surveys

  The dynamics of rational transforms: the topological picture.
  Russian Math. Surveys, v. 41 (1986), # 4, 43--117.

  Milnor's attractors, persistent recurrence and renormalization,
  ``Topological Methods in Modern Mathematics,
  A Symposium in Honor of John Milnor's 60th Birthday",
  Publish or Perish, 1993.

  On the borderline of real and complex dynamics.
  Proc. ICM, Z\"urich 1994. Birkh\"auser 1995, v. 2, 1203 -- 1215.

  Renormalization ideas in conformal dynamics.
  Cambridge Seminar ``Current Developments in Math.", May 1995.
  International Press, 1995. Cambridge, MA, 155 - 184.

  Six Lectures on Real and Complex Dynamics
  (based on European Lectures given in Barcelona, Copenhagen and St Petersburg in May-June 1999).

  Reviews.

  Review on the books by C. McMullen
  ``Complex dynamics and renormalization"
  and ``Renormalization and 3-manifolds which fiber over the circle".
  Bulletin of the AMS, v. 36 (1999), 103 - 107.

  Almost periodic representations

  This series of papers is joint with Yuri Lyubich.

  Spectral theory of almost periodic representations of semigroups
  Ukrainian Math. Journal, v.36 (1984), No 5, 474--478.