arXiv subjects

Eugen Mihailescu

Publications and source records attributed to Eugen Mihailescu.

At least 19 recordsLinked to original sources

arXiv2025

Local inverse measure-theoretic entropy for endomorphisms

We introduce a new notion of local inverse metric entropy along backward trajectories for ergodic measures preserved by endomorphisms (non-invertible maps) on a compact metric space. A second notion of inverse measure entropy is defined by using measurable partitions. Our notions have several useful applications. Inverse entropy can distinguish between isomorphism classes of endomorphisms on Lebesgue spaces, when they have the same forward measure-theoretic entropy. In a general setting we prove that the local inverse entropy of an ergodic measure μis equal to the forward entropy minus the folding entropy. The inverse entropy of hyperbolic measures on compact manifolds is explored, focusing on their negative Lyapunov exponents. We compute next the inverse entropy of the inverse SRB measure on a hyperbolic repellor. We prove an entropy rigidity result for special Anosov endomorphisms of \mathbb T^2, namely that they can be classified up to smooth conjugacy by knowing the entropy of their SRB measure and the inverse entropy of their inverse SRB measure. Next we study the relations between our inverse measure-theoretic entropy and the generalized topological inverse entropy on subsets of prehistories. In general we establish a Partial Variational Principle for inverse entropy. We obtain also a Full Variational Principle for inverse entropy in the case of special TA-covering maps on tori. In the end, several examples of endomorphisms are studied, such as fat baker transformations, fat solenoidal attractors, special Anosov endomorphisms, toral endomorphisms, and the local inverse entropy is computed for their SRB measures.

math.DS↗

arXiv2022

Invariant measures in non-conformal fibered systems with singularities

We study invariant measures and thermodynamic formalism for a class of endomorphisms $F_T$ which are only piecewise differentiable on countably many pieces and non-conformal. The endomorphism $F_T$ has parametrized countably generated limit sets in stable fibers. We prove a Global Volume Lemma for $F_T$ implying that the projections of equilibrium measures are exact dimensional on a non-compact global basic set $J_T$. A dimension formula for these global measures is obtained by using the Lyapunov exponents and marginal entropies. Then, we study the equilibrium measures of geometric potentials, and we prove that the dimensions of the associated measures in fibers depend real-analytically on the parameter s. Moreover, we establish a Variational Principle for dimension in fibers.

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arXiv2022

Geometry of measures in random systems with complete connections

We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and placedependent probabilities, are exact dimensional; moreover we determine their Hausdorff dimension. Next, we construct a family of fractals in the limit set of a countable IFS with overlaps S, and study the dimension for certain measures supported on these subfractals. In particular, we obtain families of measures on these subfractals which are related to the geometry of the system.

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arXiv2021

Pressures for multi-potentials in semigroup dynamics

We study several notions of topological pressure and capacities for multi-potentials $Φ\in \mathcal C(X;\mathbb R)^m$, with respect to finitely generated continuous semigroups $G$ on a compact metric space $X$. We introduce the amalgamated pressure, the condensed pressure, the trajectory pressure, the exhaustive pressure, and the respective capacities on non-compact sets $Y$, for multi-potentials $Φ$. This is done by using Carathéodory-Pesin structures. Several properties of these types of pressure, and relations between them are explored. The inverse limit of the semigroup and its relations to the above pressures are studied. These notions can be used to classify semigroup actions. We introduce a notion of measure-theoretic amalgamated entropy, and prove a Partial Variational Principle for the amalgamated pressure. Local amalgamated entropies and local exhaustive entropies are introduced for probability measures on X, and we show they provide estimates for the amalgamated and the exhaustive entropies of non-compact sets. Also we find a bound for the local exhaustive entropy for marginal measures on $X$. We apply the amalgamated pressure of the unstable multi-potential to estimate the dimensions of slices in $G$-invariant saddle-type sets.

math.DS↗

arXiv2021

Smale endomorphisms over graph-directed Markov systems

We study Smale skew product endomorphisms (introduced in [27]) now over countable graph directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply for instance to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We give also applications in ergodic number theory, for example to the continued fraction expansion, and the backward fractions expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions $\mathcal T_β$ of $β$-maps $T_β$, for arbitrary $β> 1$.

math.DS↗

arXiv2021

Thermodynamic formalism for invariant measures in iterated function systems with overlaps

We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems with overlaps $\mathcal S$. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for iterated function systems with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If $ν= π_*μ$ is a self-conformal measure, then $HD(ν) < \frac{h(μ)}{|χ(μ)|}$ if and only if the overlap number $o(\mathcal S, μ) > 1$. Examples are also discussed.

math.DS↗

arXiv2019

Pointwise dimension for a class of measures on limit sets

We study the pointwise dimension for a new class of projection measures on arbitrary fractal limit sets without separation conditions. We prove that the pointwise dimension exists a.e. for this class of measures associated to equilibrium states, and it is given by a formula in terms of Lyapunov exponents and a certain type of entropy. Thus these measures are exact dimensional. Self-conformal measures belong to the above class of measures, and this allows us to obtain a new geometric formula for their pointwise dimension. Thus for self-conformal measures we obtain also a geometric formula for their projection entropy.

math.DS↗

arXiv2019

Skew product Smale endomorphisms over countable shifts of finite type

We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabets. We describe then the thermodynamic formalism for Smale skew products over countable-to-1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We then study equilibrium states for skew products over endomorphisms generated by graph directed Markov systems, in particular for skew products over expanding Markov-Renyi(EMR) maps, and we settle the question of the exact dimensionality of such measures. In particular, this applies to skew products over the continued fractions transformation, and over parabolic maps. We prove next two results related to Diophantine approximation, which make the renowned Doeblin-Lenstra Conjecture more general and more precise, for a different class of measures than in the classical case. In the end, we prove exact dimensionality and find a computable formula for the dimension of equilibrium measures, for induced maps of natural extensions $\mathcal T_β$ of beta-maps, for arbitrary β> 1.

math.DS↗

arXiv2018

Ergodic lifts and overlap numbers

We study skew product lifts and overlap numbers for equilibrium measures μ_ψof Hölder continuous potentials ψon such lifts. We find computable formulas and estimates for the overlap numbers in several concrete significant cases of systems with overlaps. In particular we obtain iterated systems which are asymptotically irrational-to-1 and absolutely continuous on their limit sets. Then we look into the general structure of the Rokhlin conditional measures of μ_ψwith respect to different fiber partitions associated to the lift Φ, and find relations between them. Moreover we prove an estimate on the box dimension of a certain associated invariant measure ν_ψon the limit set Λby using the overlap number of μ_ψ.

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arXiv2015

Overlap functions for measures in conformal iterated function systems

We study conformal iterated function systems (IFS) $\mathcal S = \{ϕ_i\}_{i \in I}$ with arbitrary overlaps, and measures $μ$ on limit sets $Λ$, which are projections of equilibrium measures $\hat μ$ with respect to a certain lift map $Φ$ on $Σ_I^+ \times Λ$. No type of Open Set Condition is assumed. We introduce a notion of overlap function and overlap number for such a measure $\hat μ$ with respect to $\mathcal S$; and, in particular a notion of (topological) overlap number $o(\mathcal S)$. These notions take in consideration the $n$-chains between points in the limit set. We prove that $o(\mathcal S, \hat μ)$ is related to a conditional entropy of $\hat μ$ with respect to the lift $Φ$. Various types of projections to $Λ$ of invariant measures are studied. We obtain upper estimates for the Hausdorff dimension $HD(μ)$ of $μ$ on $Λ$, by using pressure functions and $o(\mathcal S, \hat μ)$. In particular, this applies to projections of Bernoulli measures on $Σ_I^+$. Next, we apply the results to Bernoulli convolutions $ν_λ$ for $λ\in (\frac 12, 1)$, which correspond to self-similar measures determined by composing, with equal probabilities, the contractions of an IFS with overlaps $\mathcal S_λ$. We prove that for all $λ\in (\frac 12, 1)$, there exists a relation between $HD(ν_λ)$ and the overlap number $o(\mathcal S_λ)$. The number $o(\mathcal S_λ)$ is approximated with integrals on $Σ_2^+$ with respect to the uniform Bernoulli measure $ν_{(\frac 12, \frac 12)}$. We also estimate $o(\mathcal S_λ)$ for certain values of $λ$.

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arXiv2015

Quantization coefficients in infinite systems

We investigate quantization coefficients for self-similar probability measures μon limit sets which are generated by systems S of infinitely many contractive similarities and by probabilistic vectors. The theory of quantization coefficients for infinite systems has significant differences from the finite case. One of these differences is the lack of finite maximal antichains, and the fact that the set of contraction ratios has zero infimum; another difference resides in the specific geometry of the non-compact limit set J of S. We prove that, for each r \in (0,1), there exists a unique positive number κ_r, so that for arbitrary κ< κ_r < κ', the κ-dimensional lower quantization coefficient of order r of μis positive, and we also give estimates for the κ'-dimensional upper quantization coefficient of order r of μ. In particular, it follows that the quantization dimension of order r of μexists, and it is equal to κ_r. The above results allow then to estimate the asymptotic errors of approximating the measure μin the L_r-Kantorovich-Wasserstein metric, with discrete measures supported on finitely many points.

math.PR↗

arXiv2015

Random countable iterated function systems with overlaps and applications

We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist many differences between our case and the finite deterministic case studied in [7], and we introduce new methods specific to the infinite and random case. We apply our results towards a problem related to a conjecture of Lyons about random continued fractions ([10]), and show that for Lebesgue-almost all parameters λ> 0, the invariant measure ν_λis exact dimensional. The finite IFS determining these continued fractions is not hyperbolic, but we can associate to it a random infinite IFS of contractions which have overlaps. We study then also other large classes of random countable iterated function systems with overlaps, namely: a) several types of random iterated function systems related to Kahane-Salem sets; and b) randomized infinite IFS in the plane which have uniformly bounded number of disc overlaps. For all the above classes, we find lower and upper estimates for the pointwise (and Hausdorff, packing) dimensions of the invariant measures.

math.DS↗

arXiv2013

A class of measures and non-stationary fractals, associated to f-expansions

We construct first a class of Moran fractals in R^d with countably many generators and non-stationary contraction rates; at each step n, the contractions depend on n-truncated sequences, and are related to asymptotic letter frequencies. In some cases the sets of contractions may be infinite at each step. We show that the Hausdorff dimension of such a fractal is equal to the zero h of a pressure function. We prove that the dimensions of these sets depend real analytically on the frequencies. Next, we apply the above construction to obtain non-stationary fractals E(x; f) \subset R^d, associated to f-expansions of real numbers x, and study the dependence of these fractals on x. We consider for instance beta-expansions, the continued fraction expansion and other f-expansions. By employing the Ergodic Theorem for invariant absolutely continuous measures and equilibrium measures, and using some probabilities for which the digits become independent random variables, we study the function x \to dim_H(E(x; f)) on the respective set of quasinormal numbers x \in [0; 1). We investigate also another class of fractals \tilde E_f (x) \subset R^d, for which both the non-stationary contraction vectors and the asymptotic frequencies depend on the f-representation of x. We obtain then some properties of the digits of x, related to \tilde E_f (x) and to equilibrium measures.

math.DS↗

arXiv2013

Upper estimates for stable dimensions of fractal sets with variable number of foldings

For a hyperbolic map f on a saddle type fractal Lambda with self-intersections, the number of f- preimages of a point x in Lambda may depend on x. This makes estimates of the stable dimensions more difficult than for diffeomorphisms or for maps which are constant-to-one. We employ the thermodynamic formalism in order to derive estimates for the stable Hausdorff dimension function delta^s on Lambda, in the case when f is conformal on local stable manifolds. These estimates are in terms of a continuous function on Lambda which bounds the preimage counting function from below. As a corollary we obtain that if delta^s attains its maximal possible value in Lambda, then the stable dimension is constant throughout Lambda, whereas the preimage counting function is constant on at least an open and dense subset of Lambda. In particular, this shows that if at some point in Lambda, the stable dimension is equal to the analogue of the similarity dimension in the stable direction at that point, then f behaves very much like a homeomorphism on Lambda. Finally we also obtain results about the stable upper box dimension for these type of fractals. We end the paper with a discussion of two explicit examples.

math.DS↗

arXiv2012

Equilibrium measures on saddle sets of holomorphic maps on P^2

We consider the case of hyperbolic basic sets $Λ$ of saddle type for holomorphic maps $f: \mathbb P^2\mathbb C \to \mathbb P^2\mathbb C$. We study equilibrium measures $μ_ϕ$ associated to a class of Hölder potentials $ϕ$ on $Λ$, and find the measures $μ_ϕ$ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $δ_{μ_ϕ}$ of $μ_ϕ$ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $μ_ϕ$ in the case when the preimage counting function is constant on $Λ$. For terminal/minimal saddle sets we prove that an invariant measure $ν$ obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the \textit{restriction} $f|_Λ$. This allows then to obtain formulas for the measure $ν$ of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of $ν$.

math.DS↗

arXiv2011

Inverse limits and statistical properties for chaotic implicitly defined economic models

In this paper we study the dynamics and ergodic theory of certain economic models which are implicitly defined. We consider 1-dimensional and 2-dimensional overlapping generations models, a cash-in-advance model, heterogeneous markets and a cobweb model with adaptive adjustment. We consider the inverse limit spaces of certain chaotic invariant fractal sets and their metric, ergodic and stability properties. The inverse limits give the set of intertemporal perfect foresight equilibria for the economic problem considered. First we show that the inverse limits of these models are stable under perturbations. We prove that the inverse limits are expansive and have specification property. We then employ utility functions on inverse limits in our case. We give two ways to rank such utility functions. First, when perturbing certain dynamical systems, we rank utility functions in terms of their \textit{average values} with respect to invariant probability measures on inverse limits, especially with respect to measures of maximal entropy. For families of certain unimodal maps we can adjust both the discount factor and the system parameters in order to obtain maximal average value of the utility. The second way to rank utility functions (for more general maps on hyperbolic sets) will be to use equilibrium measures of these utility functions on inverse limits; they optimize average values of utility functions while \textit{at the same time} keeping the disorder in the system as low as possible in the long run.

math.DS↗

arXiv2011

Entropy production and folding of the phase space in chaotic dynamics

We study the entropy production of Gibbs (equilibrium) measures for chaotic dynamical systems with folding of the phase space. The dynamical chaotic model is that generated by a hyperbolic non-invertible map $f$ on a general basic (possibly fractal) set $Λ$; the non-invertibility creates new phenomena and techniques than in the diffeomorphism case. We prove a formula for the \textit{entropy production}, involving an asymptotic logarithmic degree, with respect to the equilibrium measure $μ_ϕ$ associated to the potential $ϕ$. This formula helps us calculate the entropy production of the measure of maximal entropy of $f$. Next for hyperbolic toral endomorphisms, we prove that all Gibbs states $μ_ϕ$ have \textit{non-positive entropy production} $e_f(μ_ϕ)$. We study also the entropy production of the \textit{inverse Sinai-Ruelle-Bowen measure} $μ^-$ and show that for a large family of maps, it is \textit{strictly negative}, while at the same time the entropy production of the respective (forward) Sinai-Ruelle-Bowen measure $μ^+$ is strictly positive.

math.DS↗

arXiv2011

Isomorphism classes for certain expanding maps and their group extensions

We show that expanding toral endomorphisms, together with their respective Lebesgue measure are isomorphic to 1-sided Bernoulli shifts. This result is then extended to smooth perturbations of expanding toral endomorphisms, together with their respective measures of maximal entropy. Also we study group extensions of expanding toral endomorphisms and show that under certain, not too restrictive conditions on the extension cocycle, these skew products are 1-sided Bernoulli as well.

math.DS↗