Markov Operators and Semifractals

Lasota, Andrzej; Myjak, J.; Szarek, Tomasz

doi:10.1007/978-3-0348-7891-3_1

Cited by 9 publications

(16 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…with constant or place-dependent probabilities depending on whether all the p i are constant functions, or there exists at least one p i that is not a constant function, respectively; both the i.f.s. with constant and place-dependent probabilities have been a rich source of investigation topics: see, for example, the works by Barnsley [1], Barnsley and Demko [2], Barnsley, Demko, Elton, and Geronimo [3], Barnsley and Elton [4], Centore and Vrscay [6], Edalat [12], Jaroszewska [18], Lasota and Mackey [21], Lasota and Myjak [22][23][24][25][26], and [27], Lasota and Yorke [29], Myjak and Szarek [35], Nicol, Sidorov, and Broomhead [36], Stenflo [41][42][43][44], and [45], Szarek [46], Vrscay [48], and our papers [54][55][56], and [57].…”

Section: Introductionmentioning

confidence: 99%

An Ergodic Decomposition Defined by Transition Probabilities

Zaharopol

2008

Acta Appl Math

View full text Add to dashboard Cite

Our main goal in this paper is to prove that any transition probability P on a locally compact separable metric space (X, d) defines a Kryloff-Bogoliouboff-BeboutoffYosida (KBBY) ergodic decomposition of the state space (X, d). Our results extend and strengthen the results of Chap. 5 of Hernández-Lerma and Lasserre (Markov Chains and Invariant Probabilities, 2003) and extend our KBBY-decomposition for Markov-Feller operators that we have obtained in Chap. 2 of our monograph (Zaharopol in Invariant Probabilities of Markov-Feller Operators and Their Supports, 2005). In order to deal with the decomposition that we present in this paper, we had to overcome the fact that the LasotaYorke lemma (Theorem 1.2.4 in our book (op. cit.)) and two results of Lasota and Myjak (Proposition 1.1.7 and Corollary 1.1.8 of our work (op. cit.)) are no longer true in general in the non-Feller case.In the paper, we also obtain a "formula" for the supports of elementary measures of a fairly general type. The result is new even for Markov-Feller operators.We conclude the paper with an outline of the KBBY decomposition for a fairly large class of transition functions. The results for transition functions and transition probabilities seem to us surprisingly similar. However, as expected, the arguments needed to prove the results for transition functions are significantly more involved and are not presented here. We plan to discuss the KBBY decomposition for transition functions with full details in a small monograph that we are currently trying to write.

show abstract

Section: Introductionmentioning

confidence: 99%

An Ergodic Decomposition Defined by Transition Probabilities

Zaharopol

2008

Acta Appl Math

View full text Add to dashboard Cite

show abstract

“…This truncation of the optically thick accretion disk in DNe in quiescence was observationally invoked due to the observed time lags between the optical and UV fluxes in the rise phase of the outbursts (see reviews Lasota 2001Lasota , 2004 or due to unusual shape of the optical spectra or light curves of DNe (see e.g. Linnell et al 2005;Kuulkers et al 2011).…”

Section: Introductionmentioning

confidence: 99%

X-ray variations in the inner accretion flow of dwarf novae

Balman¹,

Revnivtsev

2012

A&A

View full text Add to dashboard Cite

Aims. We study the inner disk structure of dwarf novae (DNe; i.e., nonmagnetic cataclysmic variables). Methods. We performed power spectral analysis of the X-ray light curves obtained using the Rossi X-ray Timing Explorer (RXTE) and X-ray Multi-mirror Mission (XMM-Newton) data. We fit the power spectra with a simple model that describes variability as a result of matter fluctuations. In addition, we applied cross-correlation analysis of simultaneous UV and X-ray light curves using the XMM-Newton data to determine time lags between the different wavelength data. Results. For five DN systems, SS Cyg, VW Hyi, RU Peg, WW Cet, and T Leo we show that the UV and X-ray power spectra of their time variable light curves are similar in quiescence. All of them show a break in their power spectra, which in the framework of the model of propagating fluctuations indicates inner disk truncation. We derive the inner disk radii for these systems in a range of (10−3) × 10 9 cm. We analyze the RXTE data of SS Cyg in outburst and compare it with the power spectra, obtained during the period of quiescence. We show that during the outburst the disk moves towards the white dwarf and recedes as the outburst declines. We calculate the correlation between the simultaneous UV and X-ray light curves of the five DN studied in this work, using the XMMNewton data obtained in the quiescence and find X-ray time lags of 96−181 s. This can be explained by the travel time of matter from a truncated inner disk to the white dwarf surface. Conclusions. We suggest that, in general, DN may have truncated accretion disks in quiescence, which can also explain the UV and X-ray delays in the outburst stage and that the accretion may occur through coronal flows in the disk (e.g., rotating accretion disk coronae). Within the framework of the model of propagating fluctuations, the comparison of the X-ray/UV time lags observed by us in the case of DN systems with those detected for a magnetic intermediate polar allows us to make a rough estimate of the viscosity parameter α ∼ 0.25 in the innermost parts of the accretion flow of DN systems.

show abstract

“…Semi-attractors, sometimes called semi-fractals, are the smallest (unique) forward invariant set defined by means of the Kuratowski topological limits. We refer to [25,26] for a precise definition. Thus, these sets are also forward invariant selfsimilar sets (in particular closed sets) but in contracts with strict/pointwise attractors or quasi-attractors, semi-attractors can be non-compact.…”

Section: Invariant and Minimal Setsmentioning

confidence: 99%

On the chaos game of iterated function systems

Barrientos¹,

Ghane²,

Malicet³

et al. 2016

TMNA

View full text Add to dashboard Cite

Abstract. Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the deterministic chaos game. We also prove that every well-fibred quasi-attractor is renderable by the deterministic chaos game as well as quasi-attractors of both, symmetric and non-expansive IFSs.

show abstract

Markov Operators and Semifractals

Cited by 9 publications

References 6 publications

An Ergodic Decomposition Defined by Transition Probabilities

An Ergodic Decomposition Defined by Transition Probabilities

X-ray variations in the inner accretion flow of dwarf novae

On the chaos game of iterated function systems

Contact Info

Product

Resources

About