On a problem of Ö. Stenflo
We construct an e-chain on a locally compact space with the unique stationary distribution such that the strong law of large numbers does not hold. This answers negatively the question asked by Ö. Stenflo.
math.DS↗
arXiv subjects
Tomasz Szarek
Publications and source records attributed to Tomasz Szarek.
Proximality, stability, and central limit theorem for random maps on an interval
Stochastic dynamical systems consisting of non-invertible continuous maps on an interval are studied. It is proved that if they satisfy the recently introduced so-called $μ$-injectivity and some mild assumptions, then proximality, asymptotic stability and a central limit theorem hold.
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Unique ergodicity for noninvertible function systems on an interval
We study random dynamical systems of certain continuous functions on the unit interval. We use bounded variation to provide sufficient conditions for unique ergodicity of these systems. Several classes of examples are provided.
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Unique ergodicity for random noninvertible maps on an interval
In this short note, we investigate non-invertible stochastic dynamical systems on the unit interval $[0, 1]$. We provide a handy condition for unique ergodicity for systems that are injective in mean. On the other hand, we give concrete examples where unique ergodicity fails.
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Distortion in the group of circle homeomorphisms
Let $G$ be the group ${\rm PAff}_+({\bf S}^1)$ of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in $G$ is given.
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On uniqueness of invariant measures for random walks on HOMEO(R)
We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was already studied by Choquet and Deny (1960) in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on broader class of systems satisfying the conditions: recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$. Our work can be viewed as a subsequent paper of Babillot et al. (1997) and Deroin et al. (2013).
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Ergodicity and central limit theorem for random interval homeomorphisms
The central limit theorem for Markov chains generated by iterated function systems consisting of orientation preserving homeomorphisms of the interval is proved. We study also ergodicity of such systems.
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Generic invariant measures for iterated systems of interval homeomorphisms
It is known that Iterated Function Systems generated by orientation preserving homeomorphisms of the unit interval admit a unique invariant measure on $(0,1)$. The setup for this result is the positivity of Lyapunov exponents at both fixed points and the minimality of the induced action. With the additional requirement of continuous differentiability of maps on a fixed neighborhood of $\{0,1\}$, we present a metric in the space of such systems, which renders it complete. Using then a classical argument (and an alternative uniqueness proof), we show that almost singular invariant measures are admitted by systems lying densely in the space. This allows us to construct a residual set of systems with unique singular stationary distribution. Dichotomy between singular and absolutely continuous unique measures, is assured by taking a subspace of systems with absolutely continuous maps; the closure of this subspace is where the residual set is found. The article is dedicated to the memory of Professor Józef Myjak.
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On a Schur-like property for spaces of measures
A Banach space has the Schur property when every weakly convergent sequence converges in norm. We prove a Schur-like property for measures: if a sequence of finite signed Borel measures on a Polish space is such that it is bounded in total variation norm and such that for each bounded Lipschitz function the sequence of integrals of this function with respect to these measures converges, then the sequence converges in dual bounded Lipschitz norm or Fortet-Mourier norm to a measure. Moreover, we prove three consequences of this result: the first is equivalence of concepts of equicontinuity in the theory of Markov operators, the second is the derivation of weak sequential completeness of the space of signed Borel measures on Polish spaces from our main result and the third concerns conditions for the coincidence of weak and norm topologies on sets of measures that are bounded in total variation norm with additional properties.
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Equicontinuous Families of Markov Operators in View of Asymptotic Stability
Relation between equicontinuity, the so called e property and stability of Markov operators is studied. In particular, it is shown that any asymptotically stable Markov operator with an invariant measure such that the interior of its support is nonempty satisfies the e property.
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The Central Limit Theorem for function systems on the circle
The Central Limit Theorem for Iterated Functions Systems on the circle is proved. We study also ergodicity of such systems.
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Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
The convex set of quantum states of a composite $K \times K$ system with positive partial transpose is analysed. A version of the hit and run algorithm is used to generate a sequence of random points covering this set uniformly and an estimation for the convergence speed of the algorithm is derived. For $K\ge 3$ this algorithm works faster than sampling over the entire set of states and verifying whether the partial transpose is positive. The level density of the PPT states is shown to differ from the Marchenko-Pastur distribution, supported in [0,4] and corresponding asymptotically to the entire set of quantum states. Based on the shifted semi--circle law, describing asymptotic level density of partially transposed states, and on the level density for the Gaussian unitary ensemble with constraints for the spectrum we find an explicit form of the probability distribution supported in [0,3], which describes well the level density obtained numerically for PPT states.
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Stability of iterated function systems on the circle
We prove that any Iterated Function System of circle homeomorphisms with at least one of them having dense orbit, is asymptotically stable. The corresponding Perron-Frobenius operator is shown to satisfy the e-property, that is, for any continuous function its iterates are equicontinuous. The Strong Law of Large Numbers for trajectories starting from an arbitrary point for such function systems is also proved.
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The accessibility of convex bodies and derandomization of the hit and run algorithm
We introduce the concept of accessibility and prove that any convex body $X$ in $\mathbb R^d$ is accessible with relevant constants depending on $d$ only. This property leads to a new algorithm which may be considered as a natural derandomization of the hit and run algorithm applied to generate a sequence of random points covering $X$ uniformly. We prove stability of the Markov chain generated by the proposed algorithm and provide its rate of convergence.
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Limit Theorems for Some Markov Operators
The exponential rate of convergence and the Central Limit Theorem for some Markov operators are established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model given by Lasota & Mackey (1999).
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Law of the Iterated Logarithm for some Markov operators
The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell cycle model examined by A. Lasota and M.C. Mackey, J. Math. Biol. (1999).
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Robust Device-Independent Randomness Amplification with Few Devices
Randomness amplification is the task of transforming a source of somewhat random bits into a source of fully random bits. Although it is impossible to amplify randomness from a single source by classical means, the situation is different considering non-local correlations allowed by quantum mechanics. Here we give the first device-independent protocol for randomness amplification using a constant number of devices. The protocol involves four devices, can amplify any non-deterministic source into a fully random source, tolerates a constant rate of error, and has its correctness based solely on the assumption of no-signaling between the devices. In contrast all previous protocols either required an unbounded number of devices, or could only amplify sources sufficiently close to fully random.
quant-ph↗
The shark teeth is a topological IFS-attractor
We show that the space called shark teeth is a topological IFS-attractor, that is for every open cover of $X=\bigcup_{i=1}^nf_i(X)$, its image under every suitable large composition from the family of continuous functions $\{f_1,...,f_n\}$ lies in some set from the cover. In particular, there exists a space which is not homeomorphic to any IFS-attractor but is a topological IFS-attractor.
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