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Feliks Przytycki

Publications and source records attributed to Feliks Przytycki.

At least 19 recordsLinked to original sources
arXiv2026

There are not many periodic orbits in bunches for iteration of complex quadratic polynomials of one variable

It is proved that for every complex quadratic polynomial $f$ with Cremer's fixed point $z_0$ (or periodic orbit) for every $δ>0$, there is at most one periodic orbit of minimal period $n$ for all $n$ large enough, entirely in the disc (ball) $B(z_0, \exp -δn)$ (at most $p$ for a periodic Cremer orbit of period $p$; in fact at most one, by a renormalization remark by C.L.Petersen). Next, it is proved that the number of periodic orbits of period $n$ in a bunch $P_n$, that is for all $x,y\in P_n$, $|f^j(x)- f^j(y)|\le \exp -δn$ for all $j=0,...,n-1$, does not exceed $\exp δn$. We conclude that the geometric pressure defined with the use of periodic points coincides with the one defined with the use of preimages of an arbitrary typical point. I. Binder, K. Makarov and S. Smirnov (Duke Math. J. 2003) proved this for all polynomials but assuming all periodic orbits were hyperbolic, and asked about general situations. We prove here a positive answer for all quadratic polynomials.

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arXiv2024

On invariant measures of "satellite" infinitely renormalizable quadratic polynomials

Let f(z)=z^2+c be an infinitely renormalizable quadratic polynomial and J_\infty be the intersection of forward orbits of "small" Julia sets of its simple renormalizations. We prove that if f admits an infinite sequence of satellite renormalizations, then every invariant measure of f: J_\infty\to J_\infty is supported on the postcritical set and has zero Lyapunov exponent. Coupled with [G. Levin, F. Przytycki, W. Shen, The Lyapunov exponent of holomorphic maps. Invent. Math. 205 (2016), 363-382], this implies that the Lyapunov exponent of such f at c is equal to zero, which answers partly a question posed by Weixiao Shen.

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arXiv2021

Hausdorff and packing dimensions and measures for nonlinear transversally non-conformal thin solenoids

We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\varepsilon}$ hyperbolic, (partially) linear solenoids $Λ$ over the circle embedded in $\mathbb{R}^3$ 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 \emph{thinness}, assuming also there is an invariant $C^{1+\varepsilon}$ strong stable foliation, we prove that Hausdorff dimension ${\rm HD}(Λ\cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else ${\rm HD}(Λ)=t_0+1$. We prove also that for the packing measure $0<Π_{t_0}(Λ\cap W^s)<\infty$ but for Hausdorff measure ${\rm HM}_{t_0}(Λ\cap W^s)=0$ for all $W^s$. Also $0<Π_{1+t_0}(Λ) <\infty$ and ${\rm HM}_{1+t_0}(Λ)=0$. 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 ${\rm HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$. The latter holds due to a large deviations phenomenon.

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arXiv2021

On Hausdorff dimension of polynomial not totally disconnected Julia sets

We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1.

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arXiv2020

Thermodynamic formalism for coarse expanding dynamical systems

We consider a class of dynamical systems, which we call weakly coarse expanding, which is a generalization to the postcritically infinite case of expanding Thurston maps as discussed by Bonk-Meyer and is closely related to coarse expanding conformal systems as defined by Haissinsky-Pilgrim. We prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle. Further, if the system is defined on the 2-sphere, we prove all such results even in presence of periodic (repelling) branch points.

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arXiv2018

Thermodynamic formalism methods in one-dimensional real and complex dynamics

We survey some results on non-uniform hyperbolicity, geometric pressure and equilibrium states in one-dimensional real and complex dynamics. We present some relations with Hausdorff dimension and measures with refined gauge functions of limit sets for geometric coding trees for rational functions on the Riemann sphere. We discuss fluctuations of iterated sums of the potential $-t\log |f'|$ and of radial growth of derivative of univalent functions on the unit disc and the boundaries of range domains preserved by a holomorphic map $f$ repelling towards the domains.

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arXiv2016

Geometric pressure for multimodal maps of the interval

This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized multimodal maps, that is smooth maps f of a finite union of compact intervals in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transitive and has positive topological entropy. We prove that several notions of non-uniform hyperbolicity of f|_K are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in K hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). We prove that several definitions of geometric pressure P(t), that is pressure for the map f|_K and the potential - t log |f'|, give the same value (including pressure on periodic orbits, "tree" pressure, variational pressures and conformal pressure). Finally we prove that, provided all periodic orbits in K are hyperbolic repelling, the function P(t) is real analytic for t between the "condensation" and "freezing" parameters and that for each such t there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.

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arXiv2016

Geometric pressure in real and complex 1-dimensional dynamics via trees of preimages and via spanning sets

We consider $f:\hat I\to \R$ being a $C^3$ (or $C^2$ with bounded distortion) real-valued multimodal map with non-flat critical points, defined on $\hat I$ being the union of closed intervals, and its restriction to the maximal forward invariant subset $K\subset I$. We assume that $f|_K$ is topologically transitive. We call this setting the (generalized multimodal) real case. We consider also $f:\C\to \C$ a rational map on the Riemann sphere and its restriction to $K=J(f)$ being Julia set (the complex case). We consider topological pressure $P_{\spanning}(t)$ for the potential function $φ_t=-t\log |f'|$ for $t>0$ and iteration of $f$ defined in a standard way using $(n,\e)$-spanning sets. Despite of $ϕ_t=\infty$ at critical points of $f$, this definition makes sense (unlike the standard definition using $(n,\e)$-separated sets) and we prove that $P_{\spanning}(t)$ is equal to other pressure quantities, called for this potential {\it geometric pressure}, in the real case under mild additional assumptions, and in the complex case provided there is at most one critical point with forward trajectory accumulating in $J(f)$. $P_{\spanning}(t)$ is proved to be finite for general rational maps, but it may occur infinite in the real case. We also prove that geometric tree pressure in the real case is the same for trees rooted at all `safe' points, in particular at all points except the set of Hausdorff dimension 0.

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arXiv2015

The Lyapunov exponent of holomorphic maps

For any polynomial map with a single critical point, we prove that its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other complex dynamical systems. We prove further that for the unicritical polynomials with positive area Julia sets almost every point of the Julia set has zero Lyapunov exponent. Part of this statement generalizes as follows: every point with positive upper Lyapunov exponent in the Julia set of an arbitrary polynomial is not a Lebegue density point.

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arXiv2014

Lyapunov spectrum for multimodal maps

We study the dimension spectrum of Lyapunov exponents for multimodal maps of the interval and their generalizations. We also present related results for rational maps on the Riemann sphere.

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arXiv2010

Lyapunov spectrum for exceptional rational maps

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.

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arXiv2010

Nice inducing schemes and the thermodynamics of rational maps

We consider 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 (non-)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 of a rational map that is "expanding away from critical points" and that has arbitrarily small "nice sets" with some additional properties. Our results apply in particular to non-renormalizable polynomials without indifferent periodic points, infinitely renormalizable quadratic polynomials with a priori bounds, real quadratic polynomials, topological Collet-Eckmann rational maps, and to backward contracting rational maps. As an application, for these maps we describe the dimension spectrum of Lyapunov exponents, and of pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results.

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arXiv2006

Statistical properties of topological Collet-Eckmann maps

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'|.

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arXiv1996

Porosity of Collet-Eckmann Julia sets

We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.

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