arXiv subjects

Paweł Góra

Publications and source records attributed to Paweł Góra.

6 recordsLinked to original sources

arXiv2025

Existence of ACIM for Piecewise Expanding $C^{1+\varepsilon}$ maps

In this paper, we establish Lasota-Yorke inequality for the Frobenius-Perron Operator of a piecewise expanding $C^{1+\varepsilon}$ map of an interval. By adapting this inequality to satisfy the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem \cite{ionescu1950}, we demonstrate the existence of an absolutely continuous invariant measure (ACIM) for the map. Furthermore, we prove the quasi-compactness of the Frobenius-Perron operator induced by the map. Additionally, we explore significant properties of the system, including weak mixing and exponential decay of correlations.

math.DS↗

arXiv2024

Ulam's method for computing stationary densities of invariant measures for piecewise convex maps with countably infinite number of branches

Let $τ: I=[0, 1]\to [0, 1]$ be a piecewise convex map with countably infinite number of branches. In \cite{GIR}, the existence of absolutely continuous invariant measure (ACIM) $μ$ for $τ$ and the exactness of the system $(τ, μ)$ has been proven. In this paper, we develop an Ulam method for approximation of $f^*$, the density of ACIM $μ$. We construct a sequence $\{τ_n\}_{n=1}^\infty$ of maps $τ_n: I\to I$ s. t. $τ_n$ has a finite number of branches and the sequence $τ_n$ converges to $τ$ almost uniformly. Using supremum norms and Lasota-Yorke type inequalities, we prove the existence of ACIMs $μ_n$ for $τ_n$ with the densities $f_n$. For a fixed $n$, we apply Ulam's method with $k$ subintervals to $τ_n$ and compute approximations $f_{n,k}$ of $f_n$. We prove that $f_{n,k}\to f^*$ as $n\to \infty, k\to \infty,$ both a.e. and in $L^1$. We provide examples of piecewise convex maps $τ$ with countably infinite number of branches, their approximations $τ_n$ with finite number of branches and for increasing values of parameter $k$ show the errors $\|f^*-f_{n,k}\|_1$.

math.DS↗

arXiv2019

An Extreme Metastable Model for Quantum Measurement

Quantum experiments are observed as probability density functions. We often encounter multi-peaked densities which we model in this paper by a metastable dynamical system. The dynamics can be regarded in a thought experiment where a mouse is in either one of two disjoint traps which possess little outlets. The mouse spends most of its time in one or the other trap, and once in awhile makes its way to the other trap. Hence, a bi-peaked density function. When the experiment is observed - such as a light shone on the traps - the mouse stays in one trap. This measurement process results in a single peaked density that, we prove, can be modelled by the metastable process selecting an extreme density function, which is single peaked.

math.DS↗

arXiv2016

Statistical and Deterministic Dynamics of Maps with Memory

We consider a dynamical system to have memory if it remembers the current state as well as the state before that. The dynamics is defined as follows: $x_{n+1}=T_α(x_{n-1},x_{n})=τ(α\cdot x_{n}+(1-α)\cdot x_{n-1}),$ where $τ$ is a one-dimensional map on $I=[0,1]$ and $0<α<1$ determines how much memory is being used. $T_α$ does not define a dynamical system since it maps $U=I\times I$ into $I$. In this note we let $τ$ to be the symmetric tent map. We shall prove that for $0<α<0.46,$ the orbits of $\{x_{n}\}$ are described statistically by an absolutely continuous invariant measure (acim) in two dimensions. As $α$ approaches $0.5 $ from below, that is, as we approach a balance between the memory state and the present state, the support of the acims become thinner until at $α=0.5$, all points have period 3 or eventually possess period 3. For $0.5<α<0.75$, we have a global attractor: for all starting points in $U$ except $(0,0)$, the orbits are attracted to the fixed point $(2/3,2/3).$ At $α=0.75,$ we have slightly more complicated periodic behavior.

math.DS↗

arXiv2014

Toward A Mathematical Holographic Principle

In work started in [17] and continued in this paper our objective is to study selectors of multivalued functions which have interesting dynamical properties, such as possessing absolutely continuous invariant measures. We specify the graph of a multivalued function by means of lower and upper boundary maps $τ_{1}$ and $τ_{2}.$ On these boundary maps we define a position dependent random map $R_{p}=\{τ_{1},τ_{2};p,1-p\},$ which, at each time step, moves the point $x$ to $τ_{1}(x)$ with probability $p(x)$ and to $τ_{2}(x)$ with probability $1-p(x)$. Under general conditions, for each choice of $p$, $R_{p}$ possesses an absolutely continuous invariant measure with invariant density $f_{p}.$ Let $\boldsymbolτ$ be a selector which has invariant density function $f.$ One of our objectives is to study conditions under which $p(x)$ exists such that $R_{p}$ has $f$ as its invariant density function. When this is the case, the long term statistical dynamical behavior of a selector can be represented by the long term statistical behavior of a random map on the boundaries of $G.$ We refer to such a result as a mathematical holographic principle. We present examples and study the relationship between the invariant densities attainable by classes of selectors and the random maps based on the boundaries and show that, under certain conditions, the extreme points of the invariant densities for selectors are achieved by bang-bang random maps, that is, random maps for which $p(x)\in \{0,1\}.$

math.DS↗

arXiv2011

Instability of Isolated Spectrum for W-shaped Maps

In this note we consider $W$-shaped map $W_0=W_{s_1,s_2}$ with $\frac {1}{s_1}+\frac {1}{s_2}=1$ and show that eigenvalue 1 is not stable. We do this in a constructive way. For each perturbing map $W_a$ we show the existence of the "second" eigenvalue $λ_a$, such that $λ_a\to 1$, as $a\to 0$, which proves instability of isolated spectrum of $W_0$. At the same time, the existence of second eigenvalues close to 1 causes the maps $W_a$ behave in a metastable way. They have two almost invariant sets and the system spends long periods of consecutive iterations in each of them with infrequent jumps from one to the other.

math.DS↗