Continuity of sub-additive topological pressure with matrix cocycles *

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 567-588
Author(s):  
Rui Zou ◽  
Yongluo Cao ◽  
Yun Zhao
Keyword(s):  
Value Function ◽  
General Setting ◽  
Singular Value ◽  
Topological Pressure ◽  
Finite Collection ◽  
Pressure Function ◽  
Hölder Continuous ◽  
Affine Maps ◽  
Affine Set ◽  
Funct Anal

Abstract Let A = {A 1, A 2, …, A k } be a finite collection of contracting affine maps, the corresponding pressure function P(A, s) plays the fundamental role in the study of dimension of self-affine sets. The zero of the pressure function always give the upper bound of the dimension of a self-affine set, and is exactly the dimension of ‘typical’ self-affine sets. In this paper, we consider an expanding base dynamical system, and establish the continuity of the pressure with the singular value function of a Hölder continuous matrix cocycle. This extends Feng and Shmerkin’s result in (Feng and Shmerkin 2014 Geom. Funct. Anal. 24 1101–1128) to a general setting.

scholarly journals -spectra of measures on planar non-conformal attractors

2020 ◽  
pp. 1-19
Author(s):  
KENNETH J. FALCONER ◽  
JONATHAN M. FRASER ◽  
LAWRENCE D. LEE
Keyword(s):  
Value Function ◽  
Triangular Matrix ◽  
Iterated Function Systems ◽  
Singular Value ◽  
Lower Triangular Matrix ◽  
Pressure Function ◽  
Iterated Function ◽  
Lower Triangular ◽  
Bernoulli Measures ◽  
Nonlinear Maps

Abstract We study the $L^{q}$ -spectrum of measures in the plane generated by certain nonlinear maps. In particular, we consider attractors of iterated function systems consisting of maps whose components are $C^{1+\alpha }$ and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the $L^{q}$ -spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function.

scholarly journals Equilibrium stability for non-uniformly hyperbolic systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2619-2642 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
VANESSA RAMOS ◽  
JAQUELINE SIQUEIRA
Keyword(s):  
Hyperbolic Systems ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Equilibrium Stability ◽  
Hölder Continuous ◽  
Skew Products ◽  
Uniformly Hyperbolic Systems ◽  
Wide Family

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally, we show that these equilibrium states vary continuously in the $\text{weak}^{\ast }$ topology within such systems.

scholarly journals Self-similar and self-affine sets: measure of the intersection of two copies

2009 ◽  
Vol 30 (2) ◽  
pp. 399-440 ◽  
Author(s):  
MÁRTON ELEKES ◽  
TAMÁS KELETI ◽  
ANDRÁS MÁTHÉ
Keyword(s):  
List Type ◽  
Small Perturbations ◽  
Invariant Extension ◽  
Affine Maps ◽  
And Topology ◽  
Self Similar ◽  
Affine Set ◽  
Separation Conditions

AbstractLetK⊂ℝdbe a self-similar or self-affine set and letμbe a self-similar or self-affine measure on it. Let 𝒢 be the group of affine maps, similitudes, isometries or translations of ℝd. Under various assumptions (such as separation conditions, or the assumption that the transformations are small perturbations, or thatKis a so-called Sierpiński sponge) we prove theorems of the following types, which are closely related to each other.•(Non-stability)There exists a constantc<1 such that for everyg∈𝒢 we have eitherμ(K∩g(K))<c⋅μ(K) orK⊂g(K).•(Measure and topology)For everyg∈𝒢 we haveμ(K∩g(K))>0⟺∫K(K∩g(K))≠0̸ (where ∫Kis interior relative toK).•(Extension)The measureμhas a 𝒢-invariant extension to ℝd.Moreover, in many situations we characterize thosegfor whichμ(K∩g(K))>0. We also obtain results about thosegfor whichg(K)⊂Korg(K)⊃K.

scholarly journals Large deviation principles for non-uniformly hyperbolic rational maps

2010 ◽  
Vol 31 (2) ◽  
pp. 321-349 ◽  
Author(s):  
HENRI COMMAN ◽  
JUAN RIVERA-LETELIER
Keyword(s):  
Large Deviation ◽  
Periodic Points ◽  
Rational Maps ◽  
Pressure Function ◽  
Hölder Continuous ◽  
Large Deviation Principles ◽  
Continuous Potential ◽  
Uniform Hyperbolicity ◽  
Level 2 ◽  
Unique Equilibrium State

AbstractWe show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called ‘Topological Collet–Eckmann’. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each Hölder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.

Metric properties of some fractal sets and applications of inverse pressure

2009 ◽  
Vol 148 (3) ◽  
pp. 553-572 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Hausdorff Dimension ◽  
General Setting ◽  
Topological Pressure ◽  
Box Dimension ◽  
Stable Manifolds ◽  
Fractal Sets ◽  
Metric Properties ◽  
Basic Set ◽  
Unstable Set ◽  
Unstable Manifolds

AbstractWe consider iterations of smooth non-invertible maps on manifolds of real dimension 4, which are hyperbolic, conformal on stable manifolds and finite-to-one on basic sets. The dynamics of non-invertible maps can be very different than the one of diffeomorphisms, as was shown for example in [4,7,12,17,19], etc. In [13] we introduced a notion of inverse topological pressureP−which can be used for estimates of the stable dimension δs(x) (i.e the Hausdorff dimension of the intersection between the local stable manifoldWsr(x) and the basic set Λ,x∈ Λ). In [10] it is shown that the usual Bowen equation is not always true in the case of non-invertible maps. By using the notion of inverse pressureP−, we showed in [13] that δs(x) ≤ts(ϵ), wherets(ϵ) is the unique zero of the functiont→P−(tφs, ϵ), for φs(y):= log|Dfs(y)|,y∈ Λ and ϵ > 0 small. In this paper we prove that if Λ is not a repellor, thents(ϵ) < 2 for any ϵ > 0 small enough. In [11] we showed that a holomorphic s-hyperbolic map on2has a global unstable set with empty interior. Here we show in a more general setting than in [11], that the Hausdorff dimension of the global unstable setWu() is strictly less than 4 under some technical derivative condition. In the non-invertible case we may have (infinitely) many unstable manifolds going through a point in Λ, and the number of preimages belonging to Λ may vary. In [17], Qian and Zhang studied the case of attractors for non-invertible maps and gave a condition for a basic set to be an attractor in terms of the pressure of the unstable potential. In our case the situation is different, since the local unstable manifolds may intersect both inside and outside Λ and they do not form a foliation like the stable manifolds. We prove here that the upper box dimension ofWsr(x) ∩ Λ is less thants(ϵ) for any pointx∈ Λ. We give then an estimate of the Hausdorff dimension ofWu() by a different technique, using the Holder continuity of the unstable manifolds with respect to their prehistories.

Variation of topological pressure and dimension: from polynomials to complex Hénon maps

2013 ◽  
Vol 34 (3) ◽  
pp. 1018-1036
Author(s):  
CHRISTIAN WOLF
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Topological Pressure ◽  
Dimension Theory ◽  
Maximal Dimension ◽  
Pressure Function ◽  
Small Perturbations ◽  
Henon Maps ◽  
Hénon Maps ◽  
Regularity Results

AbstractWe study the topological pressure and dimension theory of complex Hénon maps which are small perturbations of one-dimensional polynomials. In particular, we derive regularity results for the generalized pressure function in a neighborhood of the degenerate map (i.e. the polynomial). This unifies results concerning the regularity of the pressure function for polynomials by Ruelle and for complex Hénon maps by Verjovsky and Wu. We then apply this regularity to show that the Hausdorff dimension of the Julia set is a continuous non-differentiable function in a neighborhood of the polynomial. Furthermore, we establish uniqueness of the measure of maximal dimension and show that the Hausdorff dimension of the Julia set of a complex Hénon map is discontinuous at the boundary of the hyperbolicity locus.

Lagrangian multipliers for generalized affine and generalized convex vector optimization problems of set-valued maps

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Renying Zeng
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Efficient Solutions ◽  
Vector Optimization Problems ◽  
Lagrangian Multipliers ◽  
Weakly Efficient Solutions ◽  
Affine Maps ◽  
Convex Vector Optimization ◽  
Affine Set

Abstract In this paper, we introduce some definitions of generalized affine set-valued maps: affinelike, preaffinelike, nearaffinelike, and prenearaffinelike maps. We present examples to explain that our definitions of generalized affine maps are different from each other. We derive a theorem of alternative of Farkas–Minkowski type, discuss Lagrangian multipliers for constrained set-valued optimization problems, and obtain some optimality conditions for weakly efficient solutions.

scholarly journals On the carrying dimension of occupation measures for self-affine random fields

2019 ◽  
Vol 39 (2) ◽  
pp. 459-479
Author(s):  
Peter Kern ◽  
Ercan Sönmez
Keyword(s):  
Hausdorff Dimension ◽  
Large Class ◽  
General Formula ◽  
Random Fields ◽  
Value Function ◽  
Singular Value ◽  
Occupation Measure ◽  
Occupation Measures ◽  
Alternative Approach ◽  
Path Properties

Hausdorff dimension results are a classical topic in the study of path properties of random fields. This article presents an alternative approach to Hausdorff dimension results for the sample functions of a large class of self-affine random fields. The aim is to demonstrate the following interesting relation to a series of articles by U. Zähle 1984, 1988, 1990, 1991. Under natural regularity assumptions, we prove that the Hausdorff dimension of the graph of self-affine fields coincides with the carrying dimension of the corresponding self-affine random occupation measure introduced by U. Zähle. As a remarkable consequence we obtain a general formula for the Hausdorff dimension given by means of the singular value function.

Exceptional sets for self-affine fractals

2008 ◽  
Vol 145 (3) ◽  
pp. 669-684 ◽  
Author(s):  
KENNETH FALCONER ◽  
JUN MIAO
Keyword(s):  
Hausdorff Dimension ◽  
Value Function ◽  
Singular Value ◽  
Exceptional Sets ◽  
Fourier Dimension ◽  
Almost All

AbstractUnder certain conditions the ‘singular value function’ formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

STOCHASTIC POTENTIALS OF INTERMITTENT MAPS

2020 ◽  
pp. 1-9
Author(s):  
HUAIBIN LI
Keyword(s):  
Nonempty Interior ◽  
Topological Pressure ◽  
Hölder Continuous ◽  
Continuous Potential

Consider an intermittent map  $f_{\unicode[STIX]{x1D705}}:[0,1]\rightarrow [0,1]$ and a Hölder continuous potential $\unicode[STIX]{x1D711}:[0,1]\rightarrow \mathbb{R}$ . We show that $\unicode[STIX]{x1D719}$ is stochastic for $f_{\unicode[STIX]{x1D705}}$ if and only if the topological pressure $P(f_{\unicode[STIX]{x1D705}},\unicode[STIX]{x1D711})$ satisfies $P(f_{\unicode[STIX]{x1D705}},\unicode[STIX]{x1D711})-\unicode[STIX]{x1D711}(0)>0$ . As a consequence, for each $\unicode[STIX]{x1D6FD}>0$ sufficiently small, the set of Hölder continuous potentials of exponent $\unicode[STIX]{x1D6FD}$ that are not stochastic for $f_{\unicode[STIX]{x1D705}}$ has nonempty interior in the space of all such potentials.

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