Fractal entropies and dimensions for microstates spaces

Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebraic invariant. We compute the free Hausdorff dimension in the cases where the set generates a finite-dimensional algebra or where the set consists of a single selfadjoint. We show that the Hausdorff dimension becomes additive for such sets in the presence of freeness.     

Unstable directions and fractal dimension for skew products with overlaps in fibers

A unique feature of smooth hyperbolic non-invertible maps is that of having different unstable directions corresponding to different prehistories of the same point. In this paper we construct a new class of examples of non-invertible hyperbolic skew products with thick fibers for which we prove that there exist uncountably many points in the locally maximal invariant set ?? (actually a Cantor set in each fiber), having different unstable directions corresponding to different prehistories; also we estimate the angle between such unstable directions. We discuss then the Hausdorff dimension of the fibers of ?? for these maps by employing the thickness of Cantor sets, the inverse pressure, and also by use of continuous bounds for the preimage counting function. We prove that in certain examples, there are uncountably many points in ?? with two preimages belonging to ??, as well as uncountably many points having only one preimage in ??. In the end we give examples which, also from the point of view of Hausdorff dimension, are far from being homeomorphisms on ??, as well as far from being constant-to-1 maps on ??.     

A remark on densities of hyperbolic dimensions for conformal iterated function systems with applications to conformal dynamics and fractal number theory

In this note we investigate radial limit sets of arbitrary regular conformal iterated function systems. We show that for each of these systems there exists a variety of finite hyperbolic subsystems such that the spectrum made of the Hausdorff dimensions of the limit sets of these subsystems is dense in the interval between 0 and the Hausdorff dimension of the given conformal iterated function system. This result has interesting applications in conformal dynamics and elementary fractal number theory.     

A note on uniform distribution for primes and closed orbits

In this note we study an analogue of Vinogradov’s uniform distribution result for prime numbers in the context of hyperbolic flows and their closed orbits. We obtain estimates for the Hausdorff dimension of certain exceptional sets.     

Hyperbolicity of the trace map for a strongly coupled quasiperiodic Schr?dinger operator

We consider the trace map associated with the silver ratio Schr?dinger operator as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently large. As a consequence, for this values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of this operator all coincide and are smooth functions of the coupling constant.     

The Smale horseshoes in a class of non-invertible maps in with two-direction instability

We study the existence of Smale horseshoes of new type and the uniformly hyperbolic invariant sets for a class of non-invertible maps in three-dimensional Euclidean spaces with the dimension of instability equal to two. Parameter regions are given, for which the map has a horseshoe and a uniformly hyperbolic invariant set on which the map is topologically conjugate to the two-sided fullshift on four symbols.     

Upper bound for the Hausdorff dimension of invariant sets of evolution variational inequalities

We consider the method of determining observations for obtaining an upper bound for the fractal dimension and the Hausdorff dimension of invariant sets of variational inequalities. We suggest a process for constructing determining observations, in particular, for dissipativity, with the use of frequency theorems for evolution systems (the Likhtarnikov–Yakubovich theorem). As an example, we consider a viscoelasticity problem in mechanics.     

Dimension of invariant sets for mappings with the squeezing property

We show an estimate of the fractal and Hausdorff dimension of sets invariant with respect to families of transformations. This estimate is proved under assumption that the transformations satisfy a squeezing property which is more general than the Lipschitz condition. Our results generalize the classical Moran formula [Moran PAP. Additive functions of intervals and Hausdorff measure. Proc Camb Philos Soc 1946;42:15–23].     

On upper estimates for the hausdorff dimension of negatively invariant sets of local cocycles

The paper is concerned with negatively invariant sets of local cocycles generated, in particular, by nonautonomous ordinary differential equations. Upper estimates for the Hausdorff dimension for negatively invariant sets of local cocycles are obtained using singular numbers of linearization of the cocycle and special functions of Lyapunov type.     

Analytic families of semihyperbolic generalized polynomial-like mappings

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we associate to each analytic family of semihyperbolic GPLs a weakly regular analytic family of conformal graph directed Markov systems with the Hausdorff dimension of the limit sets equal to the Hausdorff dimension of the Julia sets of the corresponding semihyperbolic GPLs.     

Analytic families of semihyperbolic generalized polynomial-like mappings

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we associate to each analytic family of semihyperbolic GPLs a weakly regular analytic family of conformal graph directed Markov systems with the Hausdorff dimension of the limit sets equal to the Hausdorff dimension of the Julia sets of the corresponding semihyperbolic GPLs.     

Collet, Eckmann and Hölder

We prove that Collet-Eckmann condition for rational functions, which requires exponential expansion only along the critical orbits, yields the H?lder regularity of Fatou components. This implies geometric regularity of Julia sets with non-hyperbolic and critically-recurrent dynamics. In particular, polynomial Collet-Eckmann Julia sets are locally connected if connected, and their Hausdorff dimension is strictly less than 2. The same is true for rational Collet-Eckmann Julia sets with at least one non-empty fully invariant Fatou component. Oblatum 22-III-1996 & 15-VII-1997     

Equilibrium states,pressure and escape for multimodal maps with holes

For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of the escape rate in the zero-hole limit.     

Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold

We prove a generalization of the well-known Douady–Oesterlé theorem on the upper bound for the Hausdorff dimension of an invariant set of a finite-dimensional mapping to the case of a smooth mapping generating a dynamical system on an infinite-dimensional Hilbert manifold. A similar estimate is given for the invariant set of a dynamical system generated by a differential equation on a Hilbert manifold. As an example, the well-known sine-Gordon equation is considered. In addition, we propose an algorithm for the Whitney stratification of semianalytic sets on finite-dimensional manifolds.     

空间离散Fitz-Hugh-Nagumo方程和双稳反应扩散方程组的渐近行为

本文利用扰动方法,研究了Ｆitz-Hugh-Ngaumo方程和双稳反应扩散方程在Neuman边值条件下空间离散后的渐近行为,证明了两个格微分方程组的不变区域、吸引集和整体吸引子的存在性,并给出了离散Ｆitz-Hugh-Ngaumo方程的整体吸引子的Hausdorff维数估计。     

Self-Conformal Multifractal Measures

A self-conformal measure is a measure invariant under a set of conformal mappings. In this paper we describe the local structure of self-conformal measures. For such a measure we divide its support into sets of fixed local dimension and give a formula for the Hausdorff and packing dimensions of these sets. Moreover, we compute the generalized dimensions of the self-conformal measure.     

Multifractal analysis of the historic set in non-uniformly hyperbolic systems

We get the exact Hausdorff dimension of the historic set for ratios of the Birkhoff average in a class of one dimensional non-uniformly hyperbolic dynamical systems.     

Iterated function systems with non-distinct fixed points

The dimension theory of self-similar sets is quite well understood in the cases when some separation conditions (open set condition or weak separation condition) or the so-called transversality condition hold. Otherwise the study of the Hausdorff dimension is far from well understood. We investigate the properties of the Hausdorff dimension of self-similar sets such that some functions in the corresponding iterated function system share the same fixed point. Then it is not possible to apply directly known techniques. In this paper we are going to calculate the Hausdorff dimension for almost all contracting parameters and calculate the proper dimensional Hausdorff measure of the attractor.     

Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group

We construct quasiconformal mappings on the Heisenberg group which change the Hausdorff dimension of Cantor-type sets in an arbitrary fashion. On the other hand, we give examples of subsets of the Heisenberg group whose Hausdorff dimension cannot be lowered by any quasiconformal mapping. For a general set of a certain Hausdorff dimension we obtain estimates of the Hausdorff dimension of the image set in terms of the magnitude of the quasiconformal distortion.     

The Hausdorff dimension of invariant measures for random dynamical systems

We consider random dynamical systems with jumps. The Hausdorff dimension of invariant measures for such systems is estimated.