Graph Directed Markov Systems

Mauldin, R. Daniel; Urbański, Mariusz

doi:10.1017/cbo9780511543050

Cited by 208 publications

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“…The corresponding results for the induced map F can be obtained using Lemma 3.1.3, Proposition 3.1.4 and Theorem 4.4.1 of [MU03]. …”

Section: Lemma 33 ([Prl07] Lemma 41) For Every Rational Map F Thementioning

confidence: 99%

“…These conditions are formulated in terms of the two variable pressure function defined in §3.4. We follow the method of [PRL07] for the construction of the conformal measures and the equilibrium states, which is based on the results of Mauldin and Urbanski in [MU03]. As in [PS08], we use a result of Zweimüller in [Zwe05] to show that the invariant measure we construct is in fact an equilibrium state.…”

Section: Strategy and Organizationmentioning

confidence: 99%

“…Finally, we use the method introduced by Stratmann and Urbanski in [SU03] to show that the pressure function is real analytic. Here we make use of the fact that the two variable pressure function is real analytic on the interior of the set where it is finite, a result shown by Mauldin and Urbanski in [MU03].…”

Section: Strategy and Organizationmentioning

confidence: 99%

“…We will apply several results of [MU03] to the induced map F. However, most of the results we need from [MU03] are stated for the associated symbolic space.…”

Section: Lemma 33 ([Prl07] Lemma 41) For Every Rational Map F Thementioning

confidence: 99%

“…see for example Lemma 2.1.1 and Lemma 2.1.2 of [MU03]. Here m is the function defined in §3.2, that to each point z ∈ D it associates the least good time of z.…”

Section: Pressure Function Of the Canonical Induced Map Let ( V V mentioning

confidence: 99%

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Nice Inducing Schemes and the Thermodynamics of Rational Maps

Przytycki

Rivera-Letelier

2010

Commun. Math. Phys.

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Abstract:We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential −t ln f , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in R, with at most two exceptions.

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“…The corresponding results for the induced map F can be obtained using Lemma 3.1.3, Proposition 3.1.4 and Theorem 4.4.1 of [MU03]. …”

Section: Lemma 33 ([Prl07] Lemma 41) For Every Rational Map F Thementioning

confidence: 99%

Section: Strategy and Organizationmentioning

confidence: 99%

Section: Strategy and Organizationmentioning

confidence: 99%

“…We will apply several results of [MU03] to the induced map F. However, most of the results we need from [MU03] are stated for the associated symbolic space.…”

Section: Lemma 33 ([Prl07] Lemma 41) For Every Rational Map F Thementioning

confidence: 99%

“…see for example Lemma 2.1.1 and Lemma 2.1.2 of [MU03]. Here m is the function defined in §3.2, that to each point z ∈ D it associates the least good time of z.…”

Section: Pressure Function Of the Canonical Induced Map Let ( V V mentioning

confidence: 99%

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Nice Inducing Schemes and the Thermodynamics of Rational Maps

Przytycki

Rivera-Letelier

2010

Commun. Math. Phys.

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A lower bound for the symbolic multifractal spectrum of a self‐similar multifractal with arbitrary overlaps

Olsen

2009

Mathematische Nachrichten

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By now the multifractal structure of self-similar measures satisfying the so-called Open Set Condition is well understood. However, if the Open Set Condition is not satisfied, then almost nothing is known. In this paper we prove a nontrivial lower bound for the symbolic multifractal spectrum of an arbitrary self-similar measure. We emphasize that we are considering arbitrary self-similar measures (and sets) which are not assumed to satisfy the Open Set Condition or similar separation conditions. Our results also have applications to self-similar sets which do not satisfy the Open Set Condition. IntroductionBy now the multifractal structure of self-similar measures satisfying the so-called Open Set Condition is well understood. However, if the Open Set Condition is not satisfied, then almost nothing is known. In this paper we provide a nontrivial lower bound for the symbolic multifractal spectrum of an arbitrary self-similar measure, see Theorem 1.1 and its corollaries in Section 1, and a nontrivial lower bound for the Rényi dimensions of an arbitrary self-similar measure, see Theorem 2.1 in Section 2. We emphasize that we are considering arbitrary self-similar measures and sets which are not assumed to satisfy any separation conditions; in particular, we are not assuming that the Open Set Condition is satisfied. As a further application of our results we obtain lower bounds for the Hausdorff dimension of self-similar sets which do not satisfy the Open Set Condition. For example, in Example 1.7 we obtain a lower bound for the Hausdorff dimension of the (0, 1, 3)-set of γ-expansions with deleted digits.

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Average distances between points in graph‐directed self‐similar fractals

Olsen

Richardson

2018

Mathematische Nachrichten

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We study several distinct notions of average distances between points belonging to graph‐directed self‐similar subsets of double-struckR. In particular, we compute the average distance with respect to graph‐directed self‐similar measures, and with respect to the normalised Hausdorff measure. As an application of our main results, we compute the average distance between two points belonging to the Drobot–Turner set TNfalse(c,mfalse) with respect to the normalised Hausdorff measure, i.e. we compute trueleft1scriptHs(TNfalse(c,mfalse))2∫TN(c,m)2false|x−yfalse|dfalse(Hs×Hsfalse)false(x,yfalse),where s denotes the Hausdorff dimension of TNfalse(c,mfalse) and Hs is the s‐dimensional Hausdorff measure; here the Drobot–Turner set (introduced by Drobot & Turner in 1989) is defined as follows, namely, for positive integers N and m and a positive real number c, the Drobot–Turner set TNfalse(c,mfalse) is the set of those real numbers x∈[0,1] for which any m consecutive base N digits in the N‐ary expansion of x sum up to at least c. For example, if N=2, m=3 and c=2, then our results show that trueleft1scriptHs(T2false(2,3false))2∫T2(2,3)2false|x−yfalse|dfalse(Hs×Hsfalse)false(x,yfalse)left126.0pt0.28em0.28em1em1em=4444λ2+2071λ+303012141λ2+5650λ+8281=0.36610656…,where λ=1.465571232… is the unique positive real number such that λ3−λ2−1=0.

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Graph Directed Markov Systems

Cited by 208 publications

References 0 publications

Nice Inducing Schemes and the Thermodynamics of Rational Maps

Nice Inducing Schemes and the Thermodynamics of Rational Maps

A lower bound for the symbolic multifractal spectrum of a self‐similar multifractal with arbitrary overlaps

Average distances between points in graph‐directed self‐similar fractals

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