Most of the papers since 1990, in their prelimenary form,
can be also downloaded from
the
IMS Preprint server
or from the
archive.
Online Book
Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I--II
ICM talk:
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
Dynamics of Schwarz reflections: survey
Mirrors of Conformal Dynamics:
Interplay between anti-rational maps, reflection groups,
Schwarz reflections, and correspondences
To appear in ``Algebraic, Complex and Arithmetic Dynamics",
Simons Symposia, eds. M. Jonsson and L. DeMarco
(2025) (joint with S. Mukherjee) .
Recent releases.
Antiholomorphic correspondences and mating II:
Shabat polynomial slices.
arXiv:2509.12357 (2025)
(joint with J. Mazor and S. Mukherjee) .
David extension of circle homeomorphisms:
welding, mating, and removability.
Memoirs AMS, v. 313 (2025), no. 1588
(joint with S. Merenkov, S. Mukherjee, and D. Ntalampekos).
Structure of hyperbolic polynomial automorphisms of C2 with
disconnected Julia set.
Proc LMS, v. 130 (2025), no 6.
(joint with R. Dujardin).
Schwarz reflections and the Tricorn.
Ann. Inst. Fourier, v.75 (2025), 1987-2100
(joint with S.-Y. Lee, N. Makarov, and S. Mukherjee).
Mating parabolic rarional maps with Hecke groups.
arXiv:2407.14780 (2024)
joint with S. Bullett, L. Lomonaco and S. Mukherjee.
Antiholomorphic correspondences and mating I: realization theorems.
Comm. AMS, v. 4 (2024), 495–547
(joint with J. Mazor and S. Mukherjee) .
Renormalization of unicritical diffeomorphisms of the disk.
arXiv: 2401.13559 (2024)
(joint with S. Crovissier, E. Pujals and J. Yang).
MLC at Feigenbaum points.
arXiv:2309.02107 (2023)
(joint with Dzmitry Dudko)
Uniform a priori bounds for neutral renormalization.
arxiv:2210.09280 (2022)
(joint with Dzmitry Dudko ).
Local connectivity of the Mandelbrot set at some satellite parameters
of bounded type.
GAFA, v.33 (2023), 912--1047
(joint with Dzmitry Dudko ).
Dynamics of Schwarz reflections: the mating phenomena.
Ann. Sci. Ec. Norm. Sup., v.56 (2023), 1825--1881
(joint with S.-Y. Lee, N. Makarov, and S. Mukherjee ).
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);
Ann. Inst. Fourier, v.75 (2025), 1987-2100
(joint with S.-Y. Lee, N. Makarov, and S. Mukherjee).
[3]
Schwarz reflections and anti-holomorphic correspondences.
arXiv:1907.09107 (2019)
Advances Math., v. 385 (2021)
(joint with S.-Y. Lee, N. Makarov, and S. Mukherjee).
[4]
On dynamical gaskets generated by rational maps, Kleinian groups,
and Schwarz reflections.
arXiv:1912.13438 (2019);
Conformal Geometry and Dynamics v.27 (2023), 1--54
(joint with R. Lodge, S. Merenkov, and S. Mukherjee ).
[5]
David extension of circle homeomorphisms:
welding, mating, and removability.
arXiv:2010.11256 (2020)
Memoirs AMS, v. 313 (2025), no. 1588
(joint with S. Merenkov, S. Mukherjee, and D. Ntalampekos).
[6]
Antiholomorphic correspondences and mating I: realization theorems.
arXiv:2303.02459 (2023)
Comm. AMS, v. 4 (2024), 495–547
(joint with J. Mazor and S. Mukherjee) .
[7]
Mirrors of Conformal Dynamics
arXiv: 2310.03316 (2023)
To appear in ``Algebraic, Complex and Arithmetic Dynamics",
Simons Symposia, eds. M. Jonsson and L. DeMarco
(2025) (joint with S. Mukherjee) .
[8]
Mating parabolic rarional maps with Hecke groups.
arXiv:2407.14780 (2024)
(joint with S. Bullett, L. Lomonaco and S. Mukherjee).
[9]
A general dynamical theory of Schwarz reflections, B-involutions,
and algebraic correspondences.
arXiv:2408.00204 (2024)
(joint with Y. Luo, M. Lyubich and S. Mukherjee. )
[10]
Antiholomorphic correspondences and mating II:
Shabat polynomial slices.
arXiv:2509.12357 (2025)
(joint with J. Mazor and S. Mukherjee) .
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.
More general unicritical dynamics
Teichmuller space of Fibonacci maps.
Regular or stochastic dynamics in real analytic families of unimodal maps.
Parapuzzle of the Multibrot set
and typical dynamics of unimodal maps.
The full renormalization horseshoe for unimodal maps of higher degree:
Exponential contraction along hybrid classes.
Fourty years of unimodal dynamics:
On the occasion Artur Avila winning the Brin prize.
Steps towards MLC
All papers in this series are joint with Jeremy Kahn.
Quasi-Additivity Law in conformal geometry.
Local connectivity of Julia sets for unictritical polynomials.
Combinatorial rigidity for unicritical polynomials.
A priori bounds for some infinitely renormalizable quadratics,
II. Decorations.
A priori bounds for some infinitely renormalizable quadratics,
III. Molecules.
Local connectivity of the Mandelbrot set at some satellite parameters
of bounded type.
MLC at Feigenbaum points.
Neutral Dynamics
Pacman renormalization and self-similarity of the Mandelbrot set
near Siegel parameters.
Uniform a priori bounds for neutral renormalization.
Complex Henon family
Polynomial diffeomorphisms of C^2, IV:
The measure of maximal entropy and laminar currents.
Distribution of periodic points of
polynomial diffeomorphisms of C^2.
Stability and bifurcations for dissipative polynomial automodphisms of C^2.
Classification of invariant Fatou components for dissipative Henon maps.
"lambda-lemma for families of Riemann surfaces and
the critical loci of complex H\'enon map"
Real Henon family.
Renormalization in the Henon family, I.
Universality but non-rigidity.
Renormalization in the Henon family, II: the heteroclinic web.
Probabilistic universality on two-dimensional dynamics.
Ising model and rational dynamics
The Julia sets and complex singularities in hierarchical Ising
models,
Lee-Yang zeros for DHL and 2D rational dynamics,
I. Foliation of the physical cylinder.
Lee-Yang zeros for DHL and 2D rational dynamics,
II. Global pluripotential interpretation.
Measure and dimension of some Julia sets and parameter sets
[1]
Typical behaviour of trajectories of a rational mapping of the sphere.
[2]
On the Lebesgue measure of the Julia set of a quadratic
polynomial.
[3]
How big is the set of infinitely renormalizable quadratics?
The volume
[4]
Hausdorff dimension and conformal measures of Feigenbaum Julia sets.
[5]
Lebesgue measure of Feigenbaum Julia sets.
Laminations
Laminations in holomorphic dynamics.
Conformal and harmonic measures on
laminations associated with rational maps.
Laminations and Holomorphic Dynamics.
A note on hyperbolic leaves and wild laminations of rational functions.
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.
[2]
Ergodicity of transitive unimodal transformations of the
interval.
[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.
[4]
Measure of solenoidal attractors of unimodal transformations of
the interval.
[5] Measure and dimension of solenoidal attractors of
one-dimensional dynamical systems.
[6] Measurable dynamics of S-unimodal maps of the interval.
[7]
Ergodic Theory for smooth one-dimensional dynamical systems.
No Wandering Intervals
Non-existence of wandering intervals and structure of topological attractors
of one-dimensional dynamical systems,
Non-existence of wandering intervals and structure of topological attractors
of one-dimensional dynamical systems,
Transcendental Dynamics
[1] Iterations of entire functions.
[2] Examples of entire functions with pathological dynamics.
[3] Dynamical properties of some classes of entire functions.
[4]
On typical behaviour of trajectories of the exponential function.
[5]
Measurable dynamics of the exponential.
[6]
Repelling periodic points and landing of rays for post-singularly bounded
exponential maps
Structural stability of rational endomorphisms
[1]
Some typical properties of the dynamics of rational maps.
[2] An analysis of stability of the dynamics of rational functions.
Entropy properties of rational endomorphisms
[1]
Entropy of analytic endomorphisms of the Riemann sphere.
[2]
The measure of maximal entropy of a rational endomorphism of the Riemann sphere.
[3] Entropy properties of rational endomorphisms of the Riemann sphere.
Various
On cycles and coverings associated to a knot.
Quasisymmetries of Sierpinski carpet Julia sets
Surveys
The dynamics of rational transforms: the topological picture.
Milnor's attractors, persistent recurrence and renormalization,
On the borderline of real and complex dynamics.
Renormalization ideas in conformal dynamics.
Six Lectures on Real and Complex Dynamics
Reviews.
Review on the books by C. McMullen
Almost periodic representations
This series of papers is joint with Yuri Lyubich.
Spectral theory of almost periodic representations of semigroups
Preprint IMS at Stony Brook, 1993, # 12.
Inventiones Math., v. 154 (2003), 451 -- 550
(joint with A. Avila and W. de Melo ).
Journal of European Math Soc., v. 13 (2011), 27--56.
(joint with A. Avila and W. Shen ).
Publications IHES, 114 (2012), p. 171--223
(joint with Artur Avila).
Journal of Modern Dynamics., v. 6 (2012), 183--203.
Annals Math., v. 169 (2009), 561--593.
(joint with Jeremy Kahn ).
Annals of Math., v. 170 (2009), 413--426.
(joint with Jeremy Kahn ).
Annals of Math., v. 170 (2009), 783--797
(joint with A. Avila and W. Shen).
(joint with Jeremy Kahn ).
Ann. Sci. Ecole Norm. Sup., v. 41 (2008), 57--84.
(joint with Jeremy Kahn ).
In: ``Complex Dynamics: Families and Friends''.
Proceeding of the conference dedicated to Hubbard's 60th birthday
(ed.: D. Schleicher ). Peters, A K, Limited, 2009.
(joint with Jeremy Kahn ).
GAFA, v.33 (2023), 912--1047
(joint with Dzmitry Dudko ).
arXiv:2309.02107 (2023)
(joint with Dzmitry Dudko)
JAMS, v. 33 (2020), 653--733
(joint with Dzmitry Dudko and Nikita Selinger).
arxiv:2210.09280 (2022)
(joint with Dzmitry Dudko ).
Inventions Math., v. 112 (1993), 77 - 125
(joint with Eric Bedford and John Smillie).
Inventions Math., v. 114 (1993), 277 - 288
(joint with E. Bedford and J. Smillie) .
Inventiones Math., v. 200 (2015), 439--511
(joint with Romain Dujardin).
GAFA, v. 24 (2014), 887--915
(joint with Han Peters).
Preprint IMS at Stony Brook, #3 (2014)
(joint with Tanya Firsova ).
J. Stat.Phys., Special issue dedicated to Feigenbaum's 60th birthday,
v. 121 (2005), 611--669
(joint with A. de Carvalho and M. Martens).
Inventiones Math., v. 186 (2011), 115--189.
(joint with Marco Martens).
Preprint IMS at Stony Brook, # 2 (2011)
(joint with M. Martens).
Commun. Math. Phys. v. 141 (1992), 453--474.
(joint with Pavel Bleher)
Preprint IMS at Stony Brook 2010 # 4
(joint with P. Bleher and R. Roeder )
Preprint IMS at Stony Brook 2011 # 3
(joint with P. Bleher and R. Roeder )
Dokl. Akad. Nauk SSSR, v. 268 (1982), 29 - 32.
Preprint IMS at Stony Brook, 1991, # 10.
"Voronezh Winter Mathematical Schools"
in honor of 80th birthday of S.G. Krein.
AMS Transl. (2), v. 184 (1998), 131 - 143.
J. of the AMS, 21 (2008), 305--383 (joint with A. Avila ).
arXiv: 1504.02986 [mathDS] (2015) (joint with Artur Avila ).
J. Diff. Geom., v. 47 (1997), 17 - 94 (joint with Yair Minsky).
Memoirs of the AMS, v. 173 (2005), No 820
(joint with Vadim Kaimanovich).
Lecture Notes of the mini-course given at the Conference
``New Directions in Dynamical Systems" in Kyoto, August 2002.
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).
Functional Analysis and Appl., v. 21 (1987), 70 - 71.
Ukrainian Math. J., v. 41 (1989), No 7, 985 - 988
Algebra and Analysis, v.1 (1989), 128 - 145.
English translation: Leningrad Math. J., v. 1 (1989), 137 - 155.
Math. Notes., v. 48 (1990), No 5, 15 - 20.
Comm. Math. Phys., v. 127 (1990), 573-583.
Annalles Scientifique Ecole Normale Sup., v. 24 (1991), 545-573.
Preprint IMS at Stony Brook, 1991, No 11.
I. The case of negative Schwarzian derivative.
Ergodic Theory and Dynamical Systems, v. 9 (1989), No 4, 737-750.
II. The smooth case.
Ergodic Theory and Dynamical Systems,
v. 9 (1989), No 4, 751 - 758 (joint with A. Blokh).
Dokl. Akad. Nauk SSSR, v. 279 (1984), No 1, 25-27
(joint with Alex Eremenko).
J. London Math. Soc., v. 36 (1987), 458 - 468
(joint with Alex Eremenko).
Ann. Inst. Fourier, v. 42 (1992), No 4, 989-1020
(joint with Alex Eremenko).
Russian Math, Surveys, v. 41 (1986), 199 - 200.
Siberian Math. J., v. 28 (1987), No 5, 111 - 127.
Ann. Inst Fourier, v. 64 (2014), 1493--1520
(joint with Anna Benini).
Russian Math. Surveys, v. 38 (1983), No 5, 154-155.
Teoriya Funk., Funk. Anal. and Prilozh., 42 (1984), 72 - 91
(Russian).
English translation: Selecta Mathematica Sovetica, v. 9 (1990),
69 - 90.
Functional Analysis & Appl., v. 15 (1981), No 4, 83-84.
Functional Analysis and Appl., v. 16 (1982), No 4, 78 - 79.
Ergodic Theory & Dynamical Systems, v. 3 (1983), No 3, 351-385.
On the logarithmic property of the degree of a finite group.
Dokl. Akad. Nauk SSSR, v. 247 (1979), 791--794.
arXiv math 1 301.2205v1 (2013)
(joint with Lilya Lyubich).
Preprint IMS at Stony Brook, #1 (2014)
(joint with Mario Bonk and Sergei Merenkov> )
Russian Math. Surveys, v. 41 (1986), # 4, 43--117.
``Topological Methods in Modern Mathematics,
A Symposium in Honor of John Milnor's 60th Birthday",
Publish or Perish, 1993.
Proc. ICM, Z\"urich 1994. Birkh\"auser 1995, v. 2, 1203 -- 1215.
Cambridge Seminar ``Current Developments in Math.", May 1995.
International Press, 1995. Cambridge, MA, 155 - 184.
(based on European Lectures given in Barcelona, Copenhagen and St Petersburg in May-June 1999).
``Complex dynamics and renormalization"
and ``Renormalization and 3-manifolds which fiber over the
circle".
Bulletin of the AMS, v. 36 (1999), 103 - 107.
Ukrainian Math. Journal, v.36 (1984), No 5, 474--478.