arXiv subjects

Davor Dragicevic

Publications and source records attributed to Davor Dragicevic.

At least 19 recordsLinked to original sources

arXiv2026

Decay of correlations and limit theorems for random intermittent maps

In this paper, we revisit the problem of polynomial memory loss and the central limit theorem for time-dependent LSV maps. More precisely, we show that for random LSV maps corresponding to a random parameter beta() we obtain quenched memory loss, decay of correlations, central limit theorems with rates, moment bounds and almost sure invariance principles (ASIP) when the essential infimum of beta() is less than 1/5 and the driving process (i.e. random environment) is mixing sufficiently fast. In [59, Corollary 3.8] the ASIP was obtained for ergodic driving systems when the essential supremum of \b{eta} is less than 1/2. As will be elaborated in Section 1, restrictions on the essential infimum are more natural in our context. Our results have an abstract form which we believe could be useful in other circumstances, as will be elaborated in a future work.

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arXiv2026

Smooth linearization of nonautonomous dynamics under general dichotomic behaviour

The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a very general form of dichotomy known as $μ$-dichotomy and that the associated $μ$-dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. We observe that our notion of $μ$-dichotomy encompasses the classical notions of exponential, polynomial and logarithmic dichotomies as very particular cases. In particular, our result is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between $μ$-dichotomy and exponential dichotomy via a suitable reparametrization of time.

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arXiv2026

Generalized dichotomies via time rescaling

For discrete-time nonautonomous linear dynamics and a large class of discrete growth rates $μ$, we show that the notion of $μ$ dichotomy (with respect to a sequence of norms) can be completely characterized in terms of ordinary and exponential dichotomy (with respect to a sequence of norms) by employing a suitable rescaling of time. Previously, such a result was known only in the particular case of polynomial dichotomies. As a nontrivial application of our results, we study the structure of a generalized Sacker-Sell spectrum and obtain a series of nonautonomous topological and smooth linearization results.

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arXiv2026

Quenched and annealed linear response for some partially hyperbolic skew products

We prove quenched and annealed statistical stability, linear response, and differentiability of asymptotic moments for parametric families of partially hyperbolic skew products, with random hyperbolic maps on the fibers. The main novelty is that the base maps also depend on the parameter, which leads to different formulas in the linear response and the derivative of the asymptotic moments with respect to the parameter. Our annealed results apply to partially hyperbolic maps that are not covered in \cite{BashCastro26,Dol,DS}.

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arXiv2026

h-Dichotomies via noncritical uniformity and expansiveness for evolution families

In a recent paper (Math. Ann. 393 (2025), 1769--1795), Elorreaga et al. have obtained a complete characterization of the notion of a $h$-dichotomy for ordinary differential equations on a finite-dimensional space in terms of the notions of $h$-expansiveness and $h$-noncriticality. Their results extended the previous results of Coppel and Palmer, which dealt with exponential dichotomies. The main objective of this note is to extend the results of Elorreaga et al. to arbitrary invertible evolution families that act on Banach spaces. We emphasize that our approach is completely different and considerably simpler from the one developed by Elorreaga et al. It is based on the time-rescaling method introduced by Dragicevic and Silva.

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arXiv2026

Effective quenched linear response for random dynamical systems

We prove ``effective'' linear response for certain classes of non-uniformly expanding random dynamical systems which are not necessarily composed in an i.i.d manner. In applications, the results are obtained for base maps with a sufficient amount of mixing. The fact that the rates are effective is then applied to obtain the differentiability of the variance in the CLT as a function of the parameter, as well as the annealed linear response. These two applications are beyond the reach of the linear response obtained in the general case, when all the random variables appearing in the bounds are only tempered. We also provide several wide examples of one-dimensional maps satisfying our conditions, as well as some higher-dimensional examples.

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arXiv2025

Robustness of nonuniform exponential dichotomies under a wider class of perturbations

The robustness property of exponential dichotomies refers to the stability of this notion under small linear perturbations. In recent work~\cite{PPX}, the authors have identified a new class of perturbations under which the notion of a nonuniform exponential dichotomy persists. In the present paper, we show that it is possible to extend this class. Moreover, unlike~\cite{PPX} where the results are restricted to the case of ordinary differential equations, in the present paper we deal with arbitrary evolution families consisting of possibly noninvertible linear operators.

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arXiv2025

Livšic regularity for random and sequential dynamics through transfer operators

We prove Livšic-type regularity results of coboundary representations for non-autonomous dynamical systems. Our results have an abstract nature and apply to several important specific situations, such as (higher-dimensional) random or sequential piecewise expanding maps and subshifts of finite type, which have applications to Markov interval maps and to finite state inhomogeneous elliptic Markov shifts, via symbolic representations. We also obtain results for some classes of non-autonomous hyperbolic systems. Our results can be seen as non-autonomous versions of a recent result obtained by Morris. However, we emphasize that our proof differs from the one mentioned previously even in the deterministic case. Finally, we show that our results provide a more relaxed characterization for having variance growth of Birkhoff sums on random and sequential dynamical systems; we show that such growth can fail only when the underlying functions are a coboundary without special restrictions on the regularity of the coboundary. For random systems, we show that this is equivalent to having a coboundary with bounded ``variation", but for sequential systems it turns out that this is no longer true, as demonstrated by examples.

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arXiv2025

Dichotomies for triangular systems on Hilbert spaces

In this article, we study the relationship between the exponential dichotomy properties of a triangular system of linear difference equations and its associated diagonal system on Hilbert spaces. We stress that all previous results in this direction were restricted to the finite-dimensional case. As in the previous work of the first two authors, we rely on the relationship between exponential dichotomies and the so-called admissibility properties. However, this approach requires nontrivial changes when passing from the finite-dimensional to the infinite-dimensional setting.

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arXiv2025

A characterization of $(μ,ν)$-dichotomies via admissibility

We present a characterization of $(μ,ν)$-dichotomies in terms of the admissibility of certain pairs of weighted spaces for nonautonomous discrete time dynamics acting on Banach spaces. Our general framework enables us to treat various settings in which no similar result has been previously obtained as well as to recover and refine several known results. We emphasize that our results hold without any bounded growth assumption and the statements make no use of Lyapunov norms. Moreover, as a consequence of our characterization, we study the robustness of $(μ, ν)$-dichotomies, i.e. we show that this notion persists under small but very general linear perturbations.

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arXiv2025

Linear response for random and sequential intermittent maps

This work establishes a quenched (trajectory-wise) linear response formula for random intermittent dynamical systems, consisting of Liverani-Saussol-Vaienti maps with varying parameters. This result complements recent annealed (averaged) results in the i.i.d setting. As an intermediate step, we show existence, uniqueness and statistical stability of the random absolutely continuous invariant probability measure (a.c.i.m.) for such non-uniformly expanding systems. Furthermore, we investigate sequential intermittent dynamical systems of this type and establish a linear response formula. Our arguments rely on the cone technique introduced by Baladi and Todd and further developed by Lepp{ä}nen. We also demonstrate that sequential systems exhibit a subtle distinction from both random and autonomous settings: they may possess infinitely many sequential absolutely continuous equivariant densities. However, only one of these corresponds to an SRB state in the sense of Ruelle.

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arXiv2025

$(L^p, L^q)$ Hyers-Ulam stability

We introduce a new concept of Hyers-Ulam stability, in which in the size of a pseudosolution of a given ordinary differential equation and its deviation from an exact solution are measured with respect to different norms. These norms are associated to $L^p$-spaces for $p\in [1, \infty]$. Our main objective is to formulate sufficient conditions under which semilinear ordinary differential equations exhibit such property. In addition, in certain special cases we obtain explicit formulas for the best Hyers-Ulam constant.

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arXiv2025

Iterated invariance principle for random dynamical systems

We prove a weak iterated invariance principle for a large class of non-uniformly expanding random dynamical systems. In addition, we give a quenched homogenization result for fast-slow systems in the case when the fast component corresponds to a uniformly expanding random system. Our techniques rely on the appropriate martingale decomposition.

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arXiv2024

Shadowing and hyperbolicity for linear delay difference equations

It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this paper, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two classes of linear delay difference equations: (a)~for nonautonomous equations with finite delays and uniformly bounded compact coefficient operators in (possibly infinite-dimensional) Banach spaces, (b)~for Volterra difference equations with infinite delay in finite dimensional spaces.

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arXiv2024

Admissibility and generalized nonuniform dichotomies for nonautonomous Random Dynamical Systems

In this paper, we introduce generalized dichotomies for nonautonomous random linear dynamical systems acting on arbitrary Banach spaces, and obtain their complete characterization in terms of an appropriate admissibility property. These generalized dichotomies are associated to growth rates satisfying mild conditions and they include the standard exponential behavior as a very particular case. As a nontrivial application, we establish the robustness property of such dichotomies under small (linear) perturbations.

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arXiv2024

Shadowing, Hyers--Ulam stability and hyperbolicity for nonautonomous linear delay differential equations

It is known that hyperbolic non\-autonomous linear delay differential equations in a finite dimensional space are Hyers--Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.

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arXiv2022

Conditional Lipschitz shadowing for ordinary differential equations

We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain non\-auto\-nomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to the state variable is uniformly negative in a neighborhood of the prescribed set. The results are applicable to important classes of model equations including the logistic equation, whose conditional shadowing has recently been studied. Several examples are constructed showing that the obtained conditions are optimal.

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arXiv2022

Smooth linearization of nonautonomous dynamics under polynomial behaviour

The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a nonuniform polynomial dichotomy and that the associated polynomial dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. This is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between polynomial and exponential dichotomies via a suitable reparametrization of time.

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