Feliks Przytycki | Polish Academy of Sciences (original) (raw)
Papers by Feliks Przytycki
Wiadomości Matematyczne, 2017
arXiv (Cornell University), Mar 19, 2020
We extend results by B. Hasselblatt, J. Schmeling in Dimension product structure of hyperbolic se... more We extend results by B. Hasselblatt, J. Schmeling in Dimension product structure of hyperbolic sets (2004), and by the third author and K. Simon in Hausdorff and packing measures for solenoids (2003), for C 1+ε hyperbolic, (partially) linear solenoids Λ over the circle embedded in R 3 non-conformally attracting in the stable discs W s direction, to nonlinear ones. Under an assumption of transversality and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing thinness, assuming also there is an invariant C 1+ε strong stable foliation, we prove that Hausdorff dimension HD(Λ ∩ W s) is the same quantity t0 for all W s and else HD(Λ) = t0 + 1. We prove also that for the packing measure 0 < Πt 0 (Λ ∩ W s) < ∞ but for Hausdorff measure HMt 0 (Λ ∩ W s) = 0 for all W s. Also 0 < Π1+t 0 (Λ) < ∞ and HM1+t 0 (Λ) = 0. A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every W s has measure HMt 0 equal to 0 and even Hausdorff dimension less than t0. The latter holds due to a large deviations phenomenon. Contents 1. Introduction. Statement of main results 2. Holonomy along unstable lamination 3. Lipschitz vs geometric measure 4. Hausdorff dimension 5. Packing measure 6. Hausdorff measure 7. Final remarks 7.1. Large deviations 7.2. Generalization to 1-dimensional expanding attractors 7.3. More on solenoids-coordinates 7.4. Summary of our strategies References
arXiv (Cornell University), Jun 26, 2008
We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed a... more We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential −t ln |f ′ |, and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in R, with at most two exceptions. Contents 1. Introduction 2 2. Preliminaries 9 3. Nice sets, pleasant couples and induced maps 13 4. From the induced map to the original map 21 5. Whitney decomposition of a pull-back 27 6. The contribution of a pull-back 31 7. Proof of the Main Theorem 35 Appendix A. Puzzles and nice couples 39 Appendix B. Rigidity, multifractal analysis, and level-1 large deviations 46 References 50
arXiv (Cornell University), Mar 19, 2006
We study geometric and statistical properties of complex rational maps satisfying the Topological... more We study geometric and statistical properties of complex rational maps satisfying the Topological Collet-Eckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and we show that this measure is exponentially mixing (it has exponential decay of correlations) and that it satisfies the Central Limit Theorem. We also show that for a complex rational map f the existence of such an invariant measure characterizes the Topological Collet-Eckmann Condition, and that this measure is the unique equilibrium state with potential −HD(J(f)) ln |f ′ |.
arXiv (Cornell University), Dec 12, 2010
We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sp... more We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sphere and characterize it by means of the Legendre-Fenchel transform of the hidden variational pressure. This pressure is defined by means of the variational principle with respect to non-atomic invariant probability measures and is associated to certain σ-finite conformal measures. This allows to extend previous results to exceptional rational maps.
Inventiones Mathematicae, 1986
Consider ddd disjoint closed subintervals of the unit interval and consider an orientation preser... more Consider ddd disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated to the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of C1C^1C1 conjugation. We solve the inverse problem posed by Dennis Sullivan: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it.
Nonlinearity, 1996
Consider d disjoint closed subintervals of the unit interval and consider an orientation preservi... more Consider d disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated with the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of 0951-7715/9/2/006/img5 conjugation. Dennis Sullivan posed the inverse problem: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it. We solve this problem in the case of finite smoothness and in the real-analytic case.
Mathematische Annalen, 2010
We study the dimension spectrum of Lyapunov exponents for rational maps on the Riemann sphere. Co... more We study the dimension spectrum of Lyapunov exponents for rational maps on the Riemann sphere. Contents 1. Introduction 1 2. Tools for non-uniformly hyperbolic dynamical systems 5 2.1. Topological pressure 5 2.2. Building bridges between unstable islands 6 2.3. Hyperbolic subsystems and approximation of pressure 9 2.4. Conformal measures 11 2.5. Hyperbolic times and conical limit points 11 3. On the completeness of the spectrum 11 4. Upper bounds for the dimension 15 4.1. No critical points in J 15 4.2. The general case 19 5. Lower bounds for the dimension 23 5.1. The interior of the spectrum 23 5.2. The boundary of the spectrum 24 References 36
Mathematische Annalen, 1991
Israel Journal of Mathematics, 1996
We prove that for every polynomial-like holomorphic map P, if a E K (filled-in Julia set) and the... more We prove that for every polynomial-like holomorphic map P, if a E K (filled-in Julia set) and the component Ka of K containing a is either a point or a is accessible along a continuous curve from the complement of K and K~ is eventually periodic, then a is accessible along an external ray. If a is a repelling or parabolic periodic point, then the set of arguments of the external rays converging to a is a nonempty closed "rotation set", finite (if Ka is not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in C \ K. In the Appendix we discuss constructions via cutting and glueing, from P to its external map with a "hedgehog", and backward.
Israel Journal of Mathematics, 2008
In 1974 Michael Shub asked the following question [29] : When is the topological entropy of a con... more In 1974 Michael Shub asked the following question [29] : When is the topological entropy of a continuous mapping of a compact manifold into itself is estimated from below by the logarithm of the spectral radius of the linear mapping induced in the cohomologies with real coefficients? This estimate has been called the Entropy Conjecture (EC). In 1977 the second author and Micha l Misiurewicz proved [23] that EC holds for all continuous mappings of tori. Here we prove EC for all continuous mappings of compact nilmanifolds. Also generalizations for maps of some solvmanifolds and another proof via Lefschetz and Nielsen numbers, under the assumption the map is not homotopic to a fixed points free map, are provided.
Inventiones mathematicae, 2003
We show equivalence of several standard conditions for nonuniform hyperbolicity of complex ration... more We show equivalence of several standard conditions for nonuniform hyperbolicity of complex rational functions, including the Topological Collet-Eckmann condition (TCE), Uniform Hyperbolicity on Periodic orbits, Exponential Shrinking of components of pre-images of small discs, backward Collet-Eckmann condition at one point, positivity of the infimum of Lyapunov exponents of finite invariant measures on the Julia set. The condition TCE is stated in purely topological terms, so we conclude that all these conditions are invariant under topological conjugacy. For rational maps with one critical point in Julia set all the conditions above are equivalent to the usual Collet-Eckmann and backward Collet-Eckmann conditions. Thus the latter ones are invariant by topological conjugacy in the unicritical setting. We also prove that neither part of this stronger statement is valid in the multicritical case.
International Journal of Bifurcation and Chaos, 1999
Several entropy-like invariants have been defined for noninvertible maps, based on various ways o... more Several entropy-like invariants have been defined for noninvertible maps, based on various ways of measuring the dispersion of preimages and preimage sets in the past. We investigate basic properties of four such invariants, finding that their behavior in some ways differs sharply from the analogous behavior for topological entropy.
Ergodic Theory and Dynamical Systems, 1989
It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with ... more It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with negative Schwarzian derivative and positive Lyapunov exponent along the forward trajectory of the critical value with a tent map, and its inverse h−1 are Hölder continuous.
Wiadomości Matematyczne, 2017
arXiv (Cornell University), Mar 19, 2020
We extend results by B. Hasselblatt, J. Schmeling in Dimension product structure of hyperbolic se... more We extend results by B. Hasselblatt, J. Schmeling in Dimension product structure of hyperbolic sets (2004), and by the third author and K. Simon in Hausdorff and packing measures for solenoids (2003), for C 1+ε hyperbolic, (partially) linear solenoids Λ over the circle embedded in R 3 non-conformally attracting in the stable discs W s direction, to nonlinear ones. Under an assumption of transversality and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing thinness, assuming also there is an invariant C 1+ε strong stable foliation, we prove that Hausdorff dimension HD(Λ ∩ W s) is the same quantity t0 for all W s and else HD(Λ) = t0 + 1. We prove also that for the packing measure 0 < Πt 0 (Λ ∩ W s) < ∞ but for Hausdorff measure HMt 0 (Λ ∩ W s) = 0 for all W s. Also 0 < Π1+t 0 (Λ) < ∞ and HM1+t 0 (Λ) = 0. A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every W s has measure HMt 0 equal to 0 and even Hausdorff dimension less than t0. The latter holds due to a large deviations phenomenon. Contents 1. Introduction. Statement of main results 2. Holonomy along unstable lamination 3. Lipschitz vs geometric measure 4. Hausdorff dimension 5. Packing measure 6. Hausdorff measure 7. Final remarks 7.1. Large deviations 7.2. Generalization to 1-dimensional expanding attractors 7.3. More on solenoids-coordinates 7.4. Summary of our strategies References
arXiv (Cornell University), Jun 26, 2008
We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed a... more We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential −t ln |f ′ |, and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in R, with at most two exceptions. Contents 1. Introduction 2 2. Preliminaries 9 3. Nice sets, pleasant couples and induced maps 13 4. From the induced map to the original map 21 5. Whitney decomposition of a pull-back 27 6. The contribution of a pull-back 31 7. Proof of the Main Theorem 35 Appendix A. Puzzles and nice couples 39 Appendix B. Rigidity, multifractal analysis, and level-1 large deviations 46 References 50
arXiv (Cornell University), Mar 19, 2006
We study geometric and statistical properties of complex rational maps satisfying the Topological... more We study geometric and statistical properties of complex rational maps satisfying the Topological Collet-Eckmann Condition. We show that every such a rational map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and we show that this measure is exponentially mixing (it has exponential decay of correlations) and that it satisfies the Central Limit Theorem. We also show that for a complex rational map f the existence of such an invariant measure characterizes the Topological Collet-Eckmann Condition, and that this measure is the unique equilibrium state with potential −HD(J(f)) ln |f ′ |.
arXiv (Cornell University), Dec 12, 2010
We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sp... more We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sphere and characterize it by means of the Legendre-Fenchel transform of the hidden variational pressure. This pressure is defined by means of the variational principle with respect to non-atomic invariant probability measures and is associated to certain σ-finite conformal measures. This allows to extend previous results to exceptional rational maps.
Inventiones Mathematicae, 1986
Consider ddd disjoint closed subintervals of the unit interval and consider an orientation preser... more Consider ddd disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated to the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of C1C^1C1 conjugation. We solve the inverse problem posed by Dennis Sullivan: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it.
Nonlinearity, 1996
Consider d disjoint closed subintervals of the unit interval and consider an orientation preservi... more Consider d disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated with the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of 0951-7715/9/2/006/img5 conjugation. Dennis Sullivan posed the inverse problem: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it. We solve this problem in the case of finite smoothness and in the real-analytic case.
Mathematische Annalen, 2010
We study the dimension spectrum of Lyapunov exponents for rational maps on the Riemann sphere. Co... more We study the dimension spectrum of Lyapunov exponents for rational maps on the Riemann sphere. Contents 1. Introduction 1 2. Tools for non-uniformly hyperbolic dynamical systems 5 2.1. Topological pressure 5 2.2. Building bridges between unstable islands 6 2.3. Hyperbolic subsystems and approximation of pressure 9 2.4. Conformal measures 11 2.5. Hyperbolic times and conical limit points 11 3. On the completeness of the spectrum 11 4. Upper bounds for the dimension 15 4.1. No critical points in J 15 4.2. The general case 19 5. Lower bounds for the dimension 23 5.1. The interior of the spectrum 23 5.2. The boundary of the spectrum 24 References 36
Mathematische Annalen, 1991
Israel Journal of Mathematics, 1996
We prove that for every polynomial-like holomorphic map P, if a E K (filled-in Julia set) and the... more We prove that for every polynomial-like holomorphic map P, if a E K (filled-in Julia set) and the component Ka of K containing a is either a point or a is accessible along a continuous curve from the complement of K and K~ is eventually periodic, then a is accessible along an external ray. If a is a repelling or parabolic periodic point, then the set of arguments of the external rays converging to a is a nonempty closed "rotation set", finite (if Ka is not a one point) or Cantor minimal containing a pair of arguments of external rays of a critical point in C \ K. In the Appendix we discuss constructions via cutting and glueing, from P to its external map with a "hedgehog", and backward.
Israel Journal of Mathematics, 2008
In 1974 Michael Shub asked the following question [29] : When is the topological entropy of a con... more In 1974 Michael Shub asked the following question [29] : When is the topological entropy of a continuous mapping of a compact manifold into itself is estimated from below by the logarithm of the spectral radius of the linear mapping induced in the cohomologies with real coefficients? This estimate has been called the Entropy Conjecture (EC). In 1977 the second author and Micha l Misiurewicz proved [23] that EC holds for all continuous mappings of tori. Here we prove EC for all continuous mappings of compact nilmanifolds. Also generalizations for maps of some solvmanifolds and another proof via Lefschetz and Nielsen numbers, under the assumption the map is not homotopic to a fixed points free map, are provided.
Inventiones mathematicae, 2003
We show equivalence of several standard conditions for nonuniform hyperbolicity of complex ration... more We show equivalence of several standard conditions for nonuniform hyperbolicity of complex rational functions, including the Topological Collet-Eckmann condition (TCE), Uniform Hyperbolicity on Periodic orbits, Exponential Shrinking of components of pre-images of small discs, backward Collet-Eckmann condition at one point, positivity of the infimum of Lyapunov exponents of finite invariant measures on the Julia set. The condition TCE is stated in purely topological terms, so we conclude that all these conditions are invariant under topological conjugacy. For rational maps with one critical point in Julia set all the conditions above are equivalent to the usual Collet-Eckmann and backward Collet-Eckmann conditions. Thus the latter ones are invariant by topological conjugacy in the unicritical setting. We also prove that neither part of this stronger statement is valid in the multicritical case.
International Journal of Bifurcation and Chaos, 1999
Several entropy-like invariants have been defined for noninvertible maps, based on various ways o... more Several entropy-like invariants have been defined for noninvertible maps, based on various ways of measuring the dispersion of preimages and preimage sets in the past. We investigate basic properties of four such invariants, finding that their behavior in some ways differs sharply from the analogous behavior for topological entropy.
Ergodic Theory and Dynamical Systems, 1989
It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with ... more It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with negative Schwarzian derivative and positive Lyapunov exponent along the forward trajectory of the critical value with a tent map, and its inverse h−1 are Hölder continuous.