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Michal Rams

Publications and source records attributed to Michal Rams.

At least 19 recordsLinked to original sources
arXiv2025

Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms

We study nonhyperbolic and transitive partially hyperbolic diffeomorphisms having a one-dimensional center. We prove joint flexibility with respect to entropy and center Lyapunov exponent for a broad class of these systems. Flexibility means that for any given value of the center Lyapunov exponent and any value of entropy less than the supremum of entropies of ergodic measures with that exponent, there is an ergodic measure with exactly this entropy and exponent. Our hypotheses involve minimal foliations and blender-horseshoes, they formalize the interplay between two regions of the ambient space, one of center expanding and the other of center contracting type. The list of examples our results apply is rather long, a non-exhaustive list includes fibered by circles, flow-type, some Derived from Anosov diffeomorphisms, and some anomalous (non-dynamically coherent) diffeomorphisms.

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arXiv2022

Path-dependent shrinking targets in generic affine iterated function systems

We calculate the Hausdorff dimension of shrinking target sets of symbolic balls in generic affine iterated function systems with matrix norms bounded by $\tfrac 12$. Relative to earlier works of e.g. Barany and Troscheit, and Koivusalo and Ramirez, we impose only very mild condition, known as complete reducibility, on the iterated function system. We also generalise these earlier works in that the size of the target in the current work is allowed to depend on the trajectory hitting it. The formula for Hausdorff dimension we obtain is presented as a zero point of certain pressure function.

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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

Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

The increasing rate of the Birkhoff sums in the infinite iterated function systems with polynomial decay of the derivative (for example the Gauss map) is studied. For different unbounded potential functions, the Hausdorff dimensions of the sets of points whose Birkhoff sums share the same increasing rate are obtained.

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arXiv2018

Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Let $((0,1], T)$ be the doubling map in the unit interval and $φ$ be the Saint-Petersburg potential, defined by $φ(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=φ(x)+\cdots+φ(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S\_n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\{x: \lim\_{n\to\infty}S\_n(x)/n=α\} \ (α>0)$, as well as that of the set $\{x: \lim\_{n\to\infty}S\_n(x)/Ψ(n)=α\} \ (α>0)$, when $Ψ(n)=n\log n, n^a $ or $2^{n^γ}$ for $a>1$, $γ>0$. The fast increasing Birkhoff sum of the potential function $x\mapsto 1/x$ is also studied.

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arXiv2017

Birkhoff spectrum for piecewise monotone interval maps

For piecewise monotone interval maps we look at Birkhoff spectra for regular potential functions. This means considering the Hausdorff dimension of the set of points for which the Birkhoff average of the potential takes a fixed value. In the uniformly hyperbolic case we obtain complete results, in the case with parabolic behaviour we are able to describe the part of the sets where the lower Lyapunov exponent is positive. In addition we give some lower bounds on the full spectrum in this case. This is an extension of work of Hofbauer on the entropy and Lyapunov spectra.

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arXiv2017

Metrical results on the distribution of fractional parts of powers of real numbers

Denote by {$\times$} the fractional part. We establish several new metrical results on the distribution properties of the sequence ({x n }) n$\ge$1. Many of them are presented in a more general framework, in which the sequence of functions (x $\rightarrow$ x n) n$\ge$1 is replaced by a sequence (fn) n$\ge$1 , under some growth and regularity conditions on the functions fn.

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arXiv2015

Subexponentially increasing sums of partial quotients in continued fraction expansions

We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S\_n(x)=\sum\_{j=1}^n a\_j(x)$, where $x=[a\_1(x), a\_2(x), \cdots ]$ is the continued fraction expansion of an irrational $x\in (0,1)$. Precisely, for an increasing function $φ: \mathbb{N} \rightarrow \mathbb{N}$, one is interested in the Hausdorff dimension of the sets\[E\_φ= \left\{x\in (0,1): \lim\_{n\to\infty} \frac {S\_n(x)} {φ(n)} =1\right\}.\]Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case $\exp(n^γ), \ γ\in [1/2, 1)$. We show that when $γ\in [1/2, 1)$, $E\_φ$ has Hausdorff dimension $1/2$. Thus, surprisingly, the dimension has a jump from $1$ to $1/2$ at $φ(n)=\exp(n^{1/2})$. In a similar way, the distribution of the largest partial quotient is also studied.

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arXiv2015

Upper and lower fast Khintchine spectra in continued fractions

For an irrational number $x\in [0,1)$, let $x=[a\_1(x), a\_2(x),\cdots]$ be its continued fraction expansion. Let $ψ: \mathbb{N} \rightarrow \mathbb{N}$ be a function with $ψ(n)/n\to \infty$ as $n\to\infty$. The (upper, lower) fast Khintchine spectrum for $ψ$ is defined as the Hausdorff dimension of the set of numbers $x\in (0,1)$ for which the (upper, lower) limit of $\frac{1}{ψ(n)}\sum\_{j=1}^n\log a\_j(x)$ is equal to $1$. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.

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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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arXiv2013

Projections of fractal percolations

In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset \mathbb{R}^2 $ which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than 1 then the orthogonal projection to \textbf{every} line, the radial projection with \textbf{every} center, and distance set from \textbf{every} point contain intervals.

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arXiv2013

The geometry of fractal percolation

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}

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arXiv2013

The dimension of projections of fractal percolations

\emph{Fractal percolation} or \emph{Mandelbrot percolation} is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $E\subset \mathbb{R}^2$ a typical realization of the fractal percolation on the plane, {itemize} If $\dim_{\rm H}E<1$ then for \textbf{all}lines $\ell$ the orthogonal projection $E_\ell$ of $E$ to $\ell$ has the same Hausdorff dimension as $E$, If $\dim_{\rm H}E>1$ then for any smooth real valued function $f$ which is strictly increasing in both coordinates, the image $f(E)$ contains an interval. {itemize} The second statement is quite interesting considering the fact that $E$ is almost surely a Cantor set (a {\it random dust}) for a large part of the parameter domain, see \cite{Chayes1988}. Finally, we solve a related problem about the existence of an interval in the algebraic sum of $d\geq 2$ one-dimensional fractal percolations.

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arXiv2013

Multifractal analysis for expanding interval maps with infinitely many branches

In this paper we investigate multifractal decompositions based on values of Birkhoff averages of functions from a class of symbolically continuous functions. This will be done for an expanding interval map with infinitely many branches and is a generalisation of previous work for expanding maps with finitely many branches. We show that there are substantial differences between this case and the setting where the expanding map has only finitely many branches.

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arXiv2012

Inhomogeneous Diophantine approximation with general error functions

Let $\al$ be an irrational and $φ: \N \rightarrow \R^+$ be a function decreasing to zero. For any $\al$ with a given Diophantine type, we show some sharp estimations for the Hausdorff dimension of the set [E_φ(\al):={y\in \R: |n\al -y| < φ(n) \text{for infinitely many} n},] where $|\cdot|$ denotes the distance to the nearest integer.

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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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