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# workshopdec02

## Programme of Talks

First workshop of the Warwick Dynamics Symposium 2002-2003 (December 9th-13th)

### Monday 9/12

#### 10:15-11:00

Jerome Buzzi (CMAT, \'Ecole Polythechnique)

Title: Skew products of unimodal maps over weakly expanding maps

Abstract: M. Viana considered skew products of perturbations of Misiurewicz quadratic maps over strongly expanding maps. We show how the expanding condition on the basis can be weakened.

#### 11:30-12:15

Andrzej Bis' (University of Lodz)

Title: Entropy of fractals (joint work with H. Nakayama and P.Walczak)

Abstract: We consider entropy of homeomorphisms (groups and pseudogroups of homeomorphisms) of fractals. We show that any homeomorphism of the Sierpinski gasket has zero entropy but there is a pseudogroup of homeomorphisms of the Sierpinski gasket with positive entropy. The Sierpinski curve (Sierpinski carpet) admits a homeomorphism with positive entropy. Also, we study entropy of a homeomorhism of the Menger curve.

#### 1:45-2:30

Alberto A. Pinto (Universidade do Porto)

Title: Cross ratios versus smoothness

Abstract: We will present several equivalences between cross ratio distortions and degrees of smoothness of homeomorphisms in the real line. In particular, we will discuss the regularities $C^{1+Zigmund}$ and $C^{2+\alpha}$. We will give an application to the Teichmuller space of smooth conjugacies of expanding circle maps. This is a joint work with Dennis Sullivan.

#### 2:45-3:30

Flavio Ferreira (Instituto Politecnico do Porto)

Title: Hausdorff dimension bounds for the smoothness of Denjoy maps and of Plykin attractors

Abstract: We show the existence of an infinite dimensional moduli space of $C^{1+\alpha}$ Denjoy maps which are fixed points of renormalization. However, there are no $C^{1+\alpha}$ Denjoy fixed points with $\alpha$ greater than the Hausdorff dimension of the corresponding non-wandering set. This follows from our result of non-existence of Plykin attractors with holonomies smoother than one plus the Hausdorff dimension of the intersection of the unstable leaves with the attractor.

### Tuesday 10/12

#### 10:15-11:00

Alexei Tsygvinstev (Loughborough University)

Title: Herglotz function techniques in the theory of solutions of the Feigenbaum-Cvitanovic equation

Abstract: In this talk we give a short description of the Herglotz function techniques which can be successfully applied in the theory of the fixed points of the renormalization operators arising in the theory of unimodal maps. This method, initially proposed by H. Epstein, was recently enforced with help of the analytical theory of continued fractions. In particular, it is shown how one can represent solutions of the Feigenbaum-Cvitanovic equation with help of so called $g$-fractions, which are functional continued fractions of a special type. This allows us to study effectively the asymptotics of scaling parameters for period-doubling in unimodal maps with symmetric and asymmetric critical points. References 1 Continued fractions and solutions of the Feigenbaum-Cvitanovic equation, (with B.D. Mestel, A.H. Osbaldestin), C.R. Acad. Sci. Paris, t. 334, Serie I, p. 683-688, 2002 2 A priori bounds for anti-Herglotz functions with positive real parts, preprint no. 20/38, Loughborough University, 2002

#### 11:30-12:15

Krzysztof Baranski (Warsaw University)

Title: Rational-like maps

Abstract: A rational-like map is a generalization of a polynomial-like map, where instead of simply connected regions we consider finitely connected ones. The existence of the straightening theorem for such maps is connected to the classical problem of realizability of branched coverings of the sphere. We present some results in this direction.

#### 1:45-2:30

Hans Henrik Rugh (Universit\'e de Cergy-Pontoise)

Title: On conformal repellers and Bowen's formula for the Hausdorff dimension

#### 2:45-3:30

Feliks Przytycki (Polish Academy of Sciences)

Title: Pressure for holomorphic maps

Abstract: (joint work with Juan Rivera-Letelier and Stas Smirnov) I plan to sketch a proof that for all rational functions $f$ on the Riemann sphere and potential $-t\log |f'|, t\ge 0$ several notions of pressure (variational, hyperbolic, Poincar\'e's exponential divergence exponent, infimum of exponents of conformal measures) coincide. I will prove also that they coincide with the pressure defined with the use of periodic orbits under an assumption that there are not many periodic orbits with Lyapunov exponent close to 1, in particular under the topological Collet-Eckmann condition. I will briefly discuss also the case $t < 0$.

#### 4:00-4:45

Henk Bruin (University of Groeningen)

Title: Ratios of the principal nest and absolutely continous invariant measures

Abstract: The principal nest of a unimodal interval map is a sequence of neighbourhoods of the critical point that arises in a natural first return construction. In this joint work with W. Shen and S. van Strien, we show that if these neigbourhoods shrink at a sufficiently fast geometric rate, then the map has an absolutely continuous invariant probability measure. As a corollary, we obtain a strengthening of the well-known Nowicki-van Strien summability condition.

### Wednesday 11/12

#### 10:15-11:00

Stefano Luzzatto (Imperial College)

Title: The dynamics of Henon-like and Lorenz-like maps

#### 11:30-12:15

Artur Avila (Coll\`ege de France)

Title: Hausdorff dimension and the quadratic family (joint with Carlos Gustavo Moreira)

Abstract: It is well known that in the quadratic family one can find an impressive variety of statistical behavior. Examples of relatively tame behavior are given by regular (or hyperbolic) maps, infinitely renormalizable maps and stochastic maps (with an absolutely continuous invariant measure). Those three situations present stationary behavior (Birkhoff averages converge almost everywhere), and can be well described by a reasonable physical measure. But there are also pathological examples: maps with non-stationary behavior and maps with wild physical measures (for instance Gibbs measures on hyperbolic repellers, or non-ergodic physical measures). We investigate the frequency of those different statistical behaviors in parameter space. From the measure-theoretical point of view, Lyubich showed that the situation is simple: almost every parameter is either regular or stochastic. Here we adopt the point of view of Hausdorff dimension, and show that the situation is much more complicated: each of the behaviors listed above corresponds to a set of parameters with positive dimension, but we still can prove that the set of parameters which are not regular or stochastic has positive codimension.

#### 1:45-2:30

Jacek Graczyk (Orsay)

Title: Planar harmonic measure and complex dynamics

Abstract: We prove that harmonic measure on every full compact in $\CC$ is concentrated on an asymptotically porous set with positive logarithmic density (it is a standard calculation) and show that this estimate can not be generally improved. Namely, there exists (in generic sense) a locally connected Julia set of Hausdorff dimension $2$ with harmonic measure concentrated on a set which is asymptotically non-porous with positive logarithmic density. Some geometric properties of these exotic Julia sets will be also discussed.

#### 2:45-3:30

Oleg Stenkin (Imperial College)

Title: Conservative and nonconservative behaviour of reversible diffeomorphisms in Newhouse regions

Abstract: For reversible two-dimensional diffeomorphisms we establish a new type of Newhouse regions (regions of structurally instability density). We prove that in these regions there exist a dense set of diffeomorphisms having simultaneously infinitely many stable, infinitely many unstable and infinitely many elliptic type periodic orbits, as well as infinitely many stable and unstable closed invariant curves.

#### 4:00-4:45

Samuel Senti (Pennsylvania State University)

Title: Weakly expanding sets for one dimensional maps

Abstract: Jakobson's construction of absolutely continuous invariant measures for quadratic maps relies on finding a Markov partition of a neighborhood of the critical point. We study the size and the properties of the set of points that do not belong to this partition.

### Thursday 12/12

#### 10:15-11:00

Genadi Levin (Hebrew University of Jerusalem)

Title: On a limiting universality

Abstract: (Joint work with Greg Swiatek) We prove, without help from a computer, the convergence of universal unimodal maps as the order of the critical points tend to infinity. For this, we describe and study limiting real and complex dynamics.

#### 11:30-12:15

Oksana Volkova (Nat. Acad. Sci. of Ukraine)

Title: Monotonicity of topological entropy for some families of unimodal maps.

Abstract: For some special one-parameter families of unimodal maps the topological entropy is shown to be monotone increasing with respect to the parameter.

#### 1:45-2:30

Peter Ashwin (University of Exeter)

Title: Dynamics of a piecewise isometric planar map

Abstract: We report on some work (joint with Arek Goetz) on a planar piecewise isometric map with four atoms that gives coexistence of periodic regions and regions foliated with transitive interval exchange transformations. This provides an example of a PWI on a polygon that has an infinite family of periodic regions but the only aperiodic points lie on the boundary.

#### 2:45-3:30

Christopher Penrose (Queen Mary, University of London)

Title: Regular an Limit sets for Holomorphic Correspondences

Abstract: Correspondences generalise rational maps and Kleinian groups. We consider behaviour of orbits under forward, backward and mixed iteration of a correspondence. The regular set of a correspondence is defined as the set of points having a neighbourhood whose set of return branches is accounted for by finitely many iterates. The normality set is defined as the set of points having a neighbourhood with a finite ramified cover from which all branches os iteration resolve to single-valued maps and for which the resulting collection forms a normal family. It is known that the regular set is contained in the normality set and believed that its boundary is always disjoint from the normality set. Domains for which the correspondence has a hyperbolic group resolution lie in the normality set. A straightforward definition is given of an open invariant set generalising the normality set. This set is disjoint from the non-indifferent and parabolic cycles. The special case of a Bottcher domain is analysed.

#### 4:00-4:45

Mohiniso Hidirova (Moscow State University)

Title: Problems of one-dimensional maps in quantitative research of cardiac activity

Abstract: Model system of functional-differential equations of the regulation mechanisms of cardiac tissue excitation in the form of discrete equations is considered. Presence of steady trivial (0) and existence of two nontrivial ($A$, $B$, $A < B$) positions of equilibrium are shown. Nontrivial position of equilibrium $A$ is unstable, but the second ($B$) is attractor in $(A, \infty)$, which can be strange attractor. The analysis of dynamic chaos area on the basis of Lamerey diagrams, Kolmogorov entropy, Lyapunov factor, Hausdorf and high dimensionality has shown: - dynamic chaos origin in the mode of Poincare type limit cycles according to universal Feygenbaum scenario; - presence of small regions with regular fluctuations (r-windows) in the field of the dynamic chaos in parametric portrait; - individuality of quantities, sizes and locations of r-windows for different model systems; - completion of irregular fluctuations by solutions collapse in a dynamic system ('black hole' effect). Identification of the dynamic chaos area with cardiac arrhythmia and 'black hole' effect with heart death has allowed to define regularities of origin and development of arrhythmia and sudden cardiac death. The problems (quantities, sizes and locations of r-windows in the dynamic chaos area, function time of dynamic system at 'black hole') have greater importance for modelling of heart activity at norm, anomalies for recommendation development on heart rhythm correction in the field of arrhythmia and clinical cardiology. This work was supported by ¹ 41-96 FSFI AS RUz grant and ¹ 41/2000 FFI GKST RUz grant.

### Friday 13/12

#### 10:15-11:00

Andrei Sivak (Nat. Acad. Sci. of Ukraine)

Title: On relations between the dynamics and the structure of minimal attraction center of trajectories

#### 11:30-12:15

Evgueni Sataev (Obninsk Inst. of Nuclear Power Eng.)

Title: Some examples of continuous dependence of invariant measures on parameters of system.

Abstract: Systems depending on a parameter $a$, which are not hyperbolic when $a=a_0$, are considered. There exists the invariant measure of Sinai-Bowen-Ruelle type for $a\ne a_0$. This measure depends continuously on the parameter. The question is: does this measure depends continuously on the parameters at the point $a_0$? The answer is positive for some examples. The simplest example is the family of the interval $[0,1]$ defined by the equation $$f(x)=ax+x^2 ({\rm mod} 1)\quad a_0=1.$$ The second example is the perturbation of a torus automorphism for which one of the eigenvalues is equal to 1.

#### 1:45-2:30

Georgi Chakvetadze (Hebrew University of Jerusalem)

Title: Parameter random perturbations of certain quadratic maps

Abstract: Continuing the line of research carried out by Baladi, Benedicks, Katok, Kifer, Viana, Young and others we prove that many quadratic maps $F_\lambda(x)=\lambda x(1-x)$, $x\in [0,1]$, $\lambda\in [0,4]$, are strongly stochastically stable with respect to the random parameter perturbations. Given $\lambda\in [0,4]$, the random parameter perturbation of $F_\lambda$ is the family $$X_n^\varepsilon=F_{\lambda(1+\varepsilon\zeta_n)} \circ\ldots\circ F_{\lambda(1+\varepsilon\zeta_1)}(X_0^\varepsilon), \quad n\in\bf{N},\quad \varepsilon>0, \eqno(1)$$ of Markov chains depending on small parameter $\varepsilon$, where the variables $\zeta_n$ are i.i.d. on $[-1,1]$, $X_0^\varepsilon$ is distributed on $[0,1]$ independently of $\zeta_n$, $n\in\bf{N}$. 'Strong stochastic stability' means that for any small enough $\varepsilon$ there is a unique absolutely continuous stationary distribution of the chain $\{X_n^\varepsilon\}$ with density converging in $\bf{L}_1$-metric to the unique normalized invariant density of $F_\lambda$ (which also exists) as $\varepsilon$ tends to zero. 'Many' means that the stability assertion has been proved for a positive Lebesgue measure set of $\lambda's$.

#### 2:45-3:30

Konstantin Khanin (University of Cambridge)

Title: Rigidity for circle homeomorphisms.