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Ivan Werner

Publications and source records attributed to Ivan Werner.

12 recordsLinked to original sources
arXiv2020

Erratum: Coding map for a contractive Markov system

An error in the proof of Lemma 2 (ii) in [I. Werner, Math. Proc. Camb. Phil. Soc. 140(2) 333-347 (2006)], which claims the absolute continuity of dynamically defined measures (DDM), is identified. This undermines the assertion of the positivity of a DDM which provides a construction for equilibrium states in [I. Werner, J. Math. Phys. 52 122701 (2011)]. An explicit lower bound for the DDM appearing there is computed in the case when all maps of a contractive Markov system (CMS) are contractions, the probability functions are Dini-continuous and bounded away from zero, and there exists an equilibrium state of the CMS which is absolutely continuous with respect to the initial measure. In the case of the contraction only on average, a generalized construction is shown to provide a positive set function, but it is unknown whether it gives a measure on the Borel $σ$-algebra, and if it did, the measure would coincide with the original DDM.

math.DS↗
arXiv2017

On the Carathéodory approach to the construction of a measure

The Carathéodory theorem on the construction of a measure is generalized by replacing the outer measure with an approximation of it and generalizing the Carathéodory measurability. The new theorem is applied to obtain dynamically defined measures from constructions of outer measure approximations resulting from sequences of measurement pairs consisting of refining $σ$-algebras and measures on them which need not be consistent. A particular case when the measurement pairs are given by the action of an invertible map on an initial $σ$-algebra and a measure on it is also considered.

math.FA↗
arXiv2015

Lower bounds for the dynamically defined measures

The dynamically defined measure (DDM) $\Phi$ arising from a finite measure $\phi_0$ on an initial $\sigma$-algebra on a set and an invertible map acting on the latter is considered. Several lower bounds for it are obtained and sufficient conditions for its positivity are deduced under the general assumption that there exists an invariant measure $\Lambda$ such that $\Lambda\ll\phi_0$. In particular, DDMs arising from the Hellinger integral $\mathcal{J}_\alpha(\Lambda,\phi_0)\geq\mathcal{H}^{\alpha,0}(\Lambda,\phi_0)\geq\mathcal{H}_\alpha(\Lambda,\phi_0)$ are constructed with $\mathcal{H}_{0}\left(\Lambda,\phi_0\right)(Q) = \Phi(Q)$, $\mathcal{H}_{1}\left(\Lambda,\phi_0\right)(Q) = \Lambda(Q)$, and \[\Phi(Q)^{1-\alpha}\Lambda(Q)^{\alpha}\geq\mathcal{J}_{\alpha}\left(\Lambda,\phi_0\right)(Q)\] for all measurable $Q$ and $\alpha\in[0,1]$, and further computable lower bounds for them are obtained and analyzed. The function $(0,\gamma]\owns\alpha\longmapsto\mathcal{H}_{\alpha}(\Lambda,\phi_0)$ is computed explicitly for $\gamma\geq 1$ such that $\int(d\Lambda/d\phi_0)^{\gamma-1}d\Lambda<\infty$ in the case of a discrete ergodic decomposition of $\Lambda$, and the other two functions are computed under the additional condition of the equivalence of $\phi_0$ and $\Lambda$. In particular, if $\Lambda$ is ergodic, it is shown that the first function is completely determined by the $\Lambda$-essential supremum (infimum) of $d\Lambda/d\phi_0$ for all $0<\alpha<1$ ($1<\alpha\leq\gamma$), and, if it is continuous at $0$, the above inequalities become equalities. The computation of it enables an explicit computation of some DDMs arising as outer measure approximations with respect to it, which demonstrates that this technique allows to obtain new measures, and that such measures can have phase transitions with respect to the DDM specifying the covering sets.

math.DS↗
arXiv2015

Contractive Markov systems II

Discrete time random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniform continuous and contractive are considered. A notion of a generating communication class of such a system is introduced, which includes every communication class if the system has a finite Markov partition. It is shown that the ergodic decomposition of an equilibrium state associated with such a system is purely atomic and can be exhaustively described using the generating communication classes if the system satisfies an absolute continuity condition (ACC). In such a case, each invariant Borel probability measure which is an image of an ergodic component of an equilibrium state under the coding map can be obtained by a random walk starting at any point in the corresponding generating communication class. As a by-product, a practical method for a computation of the entropy of the equilibrium states is obtained. Finally, it is shown that such a non-degenerate system satisfying the ACC which in addition has a dominating Markov chain and a finite (20) has a unique invariant Borel probability measure if and only if it has a single generating communication class. Some sufficient conditions for the ACC are provided.

math.PR↗
arXiv2015

Dynamically defined measures and equilibrium states

A technique of dynamically defined measures is developed and its relation to the theory of equilibrium states is shown. The technique uses Caratheodory's method and the outer measure introduced in (I. Werner, Math. Proc. Camb. Phil. Soc. 140 (2) (2006) 333-347). As an application, equilibrium states for contractive Markov systems (I. Werner, J. London Math. Soc. 71 (2005), no. 1, 236-258) are obtained.

math-ph↗
arXiv2014

Equilibrium states and invariant measures for random dynamical systems

Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient condition for the existence of an invariant Borel probability measure for such a non-degenerate system with a dominating Markov chain and a finite (16) is given. The condition is also sufficient if the non-degeneracy is weakened with the consistency condition. A further sufficient condition for the existence of an invariant measure for such a consistent system which involves only the properties of the dominating Markov chain is provided. In particular, it implies that every such a consistent system with a finite Markov partition and a finite (16) has an invariant Borel probability measure. A bijective map between these measures and equilibrium states associated with such a system is established in the non-degenerate case. Some properties of the map and the measures are given.

math.DS↗
arXiv2012

On coding with Feller contractive Markov systems

We continue development of the theory of contractive Markov systems initiated in \cite{Wer1}. In this paper, we construct the coding map for Feller contractive Markov systems. This allows us to prove a generalization of Ledrappier's Theorem \cite{Le} and to show the existence of observable invariant measures for Feller contractive Markov system.

math.DS↗
arXiv2009

Fundamental Markov systems

We continue development of the theory of Markov systems initiated in \cite{Wer1}. In this paper, we introduce fundamental Markov systems associated with random dynamical systems and show that the proof of the uniqueness and empiricalness of the stationary initial distribution of the random dynamical system reduces to that for the fundamental Markov system associated with it. The stability criteria for the latter are much clearer.

math.PR↗
arXiv2005

Kolmogorov-Sinai entropy of a generalized Markov shift

In this paper we calculate Kolmogorov-Sinai entropy $h_M(S)$ of the generalized Markov shift associated with a contractive Markov system (CMS) \cite{Wer1} using the coding map constructed in \cite{Wer3}. We show that \[h_M(S)=-\sum\limits_{e\in E}\int\limits_{K_{i(e)}} p_e\log p_edμ\] where $μ$ is a unique invariant Borel probability measure of the CMS. I. Werner, Contractive Markov systems, J. London Math. Soc. (2005) 236-258. I. Werner, Coding map for a contractive Markov system, Math. Proc. Camb. Phil. Soc. to appear 140 (2), March 2006.

math.DS↗
arXiv2005

The generalized Markov measure as an equilibrium state

In this paper, we continue development of the theory of contractive Markov systems (CMS) initiated in \cite{Wer1}. Also, this work can be seen as a small contribution to the theory of equilibrium states. We construct an energy function on the code space, using the coding map from \cite{Wer3}, and show that the generalized Markov measure associated with an irreducible CMS is a unique equilibrium state for this energy function if the vertex sets form an open partition of the state space of the CMS and the restrictions of the probability functions on their vertex sets are Dini-continuous and bounded away from zero.

math.DS↗
arXiv2005

Coding map for a contractive Markov system

In this paper, we develop the theory of contractive Markov systems initiated in \cite{Wer1}. We construct a coding map for such systems and investigate some of its properties.

math.PR↗