Abstract
The aim of this paper is to find an upper bound for the box-counting dimension of uniform attractors for non-autonomous dynamical systems. Contrary to the results in the literature, we do not require the symbol space to have finite box-counting dimension. Instead, we ask a condition on the semi-continuity of pullback attractors of the system as time goes to infinity. This semi-continuity can be achieved if we suppose the existence of finite-dimensional exponential uniform attractors for the limit symbols, that is, the symbols associated with the asymptotic vector fields of the non-autonomous differential equation. After showing these new results, we apply them to study the box-counting dimension of the uniform attractor for a non-autonomous reaction-diffusion equation, and we find a class of forcing terms for this equation such that their associated symbol spaces have infinite box-counting dimension, but the uniform attractors of the dynamical systems generated by these forcing terms have finite box-counting dimension anyway.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Data Availability
No datasets were generated or analyzed during the current study.
Notes
The Hausdorff semi-distance \(\text{ dist}_H(A,B)\) shows how close an arbitrary point of A is to some point of B. If \(\text{ dist}_H(A,B) < \epsilon \), then the set A is in an \(\epsilon \) neighborhood of the set B. Note that the Hausdorff semi-distance is generally not symmetric; it measures closeness in only one direction.
See Ott and Yorke (2005). A prevalent set is a measure-theoretic analog of “almost every” in infinite-dimensional spaces. In particular, prevalent sets are dense, and the countable intersection of prevalent sets is also prevalent.
If we have a pair of Banach spaces \(Y\subset X\) with Y compactly embedded in X, the Kolmogorov \(\epsilon \)-entropy (see Triebel (1978)) of this embedding is defined as \(\varvec{H}_{\epsilon }(Y;X) = \log _2 N_{\epsilon }\), where \(N_{\epsilon }\) is the minimal number of balls of radius \(\epsilon \) in X necessary to cover the unit ball \(B_Y(0,1)\) in Y. This represents the fractal complexity of the set \(B_Y(0,1)\) in X.
References
Amann, H.: Linear and Quasilinear Parabolic Problems. Birkhäuser Basel, New York (1995)
Arrieta, J.M., Carvalho, A.N., Rodríguez-Bernal, A.: Attractors of parabolic problems with nonlinear boundary conditions. Uniform bounds. Commun. Partial Diff. Equ. 25, 1–37 (2000)
Babin, A. V., Vishik, M. I.: Attractors of Evolution Equations. Studies in Mathematics and its Applications, 25. North-Holland, Amsterdam, (1992)
Carvalho, A. N., Bortolan, M. C., and Langa, J.: Attractors Under Autonomous and Non-autonomous Perturbations. American Mathematical Society, 05 (2020)
Carvalho, A.N., Cholewa, J.W.: Strongly damped wave equations in \( w_0^{1, p}(\omega ) \times l^p(\omega ) \). Discrete Contin. Dyn. Syst. - Suppl. 2007, 230–239 (2007)
Carvalho, A.N., Langa, J., Robinson, J.C.: Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems. Springer, New-York (2013)
Carvalho, A.N., Oliveira-Sousa, A.N.: Navier-stokes equations: a millennium prize problem from the point of view of continuation of solutions. Matemática Contemporânea 59, 4–17 (2024)
Carvalho, A.N., Sonner, S.: Pullback exponential attractors for evolution processes in banach spaces: theoretical results. Commun. Pure Appl. Anal. 12(6), 3047–3071 (2013)
Chepyzhov, V., Vishik, M.: A hausdorff dimension estimate for kernel sections of non-autonomous evolution equations. Indiana Univ. Math. J. 42(3), 1057–1076 (1993)
Chepyzhov, V., Vishik, M.: Attractors for Equations of Mathematical Physics. American Mathematical Society, Vol. 49 (2002)
Cholewa, J., Czaja, R., and Mola, A.: Remarks on the fractal dimension of bi-space global and exponential attractors. Bollettino della Unione Matematica Italiana. Series IX 1 (2008)
Conway, J.B.: Functions of One Complex Variable I. Springer-Verlag, New York (1978)
Cui, H., Carvalho, A.N., Cunha, A.C., Langa, J.A.: Smoothing and finite-dimensionality of uniform attractors in banach spaces. J. Diff. Equ. 285, 383–428 (2021)
Cui, H., Figueroa-López, R.N., Langa, J.A., Nascimento, M.J.D.: Forward attraction of nonautonomous dynamical systems and applications to navier-stokes equations. SIAM J. Appl. Dyn. Syst. 23(3), 2407–2443 (2024)
Efendiev, M., Miranville, A., Zelik, S.: Global and exponential attractors for nonlinear reaction-diffusion systems in unbounded domains. Proc. Royal Soc.Edinburgh: Sect. A Math. 134, 271–315 (2004)
Foias, C., Olson, E.: Finite fractal dimension and Hölder-Lipschitz parametrization. Indiana Univ. Math. J. 45(3), 603–616 (1996)
Hunt, B.R., Kaloshin, V.: Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity 12, 1263–1275 (1999)
Kloeden, P., Lorenz, T.: Pullback and forward attractors of nonautonomous difference equations. In Difference Equations, Discrete Dynamical Systems and Applications, Springer International Publishing, Cham, pp. 37–48 (2015)
Kloeden, P. E., Rasmussen, M.: Nonautonomous Dynamical Systems. Mathematical Surveys and Monographs, v. 176. American Mathematical Society, Providence, R.I, (2011)
Langa, J.A., Robinson, J.C.: A finite number of point observations which determine a non-autonomous fluid flow. Nonlinearity 14(4), 673 (2001)
Ott, W., Yorke, J.A.: Prevalence. Bull. Am. Math. Soc. 42, 263–290 (2005)
Robinson, J.C.: Linear embeddings of finite-dimensional subsets of banach spaces into euclidean spaces. Nonlinearity 22, 711–728 (2009)
Robinson, J. C.: Dimensions, Embeddings, and Attractors. Cambridge Tracts in Mathematics. Cambridge University Press, (2010)
Rosenholtz, I.: Another proof that any compact metric space is the continuous image of the cantor set. Am. Math. Mon. 83(8), 646–647 (1976)
Triebel, H.: Interpolation Theory Function Spaces. Differential Operators. North-Holland Publishing Company, Amsterdam-New York (1978)
Funding
A. N. CARVALHO.: Partially supported by FAPESP Grant # 20/14075-6 and by CNPq Grant # 308902/2023-8, Brazil. J. A. LANGA.: Partially supported by Grant # PID2021-122991NB-C21 Ministerio de Ciencia e Innovación, Spain. R. O. MOURA.: Supported by Grants #2022/04886-2 and #2023/11798-5, São Paulo Research foundation (FAPESP)
Author information
Authors and Affiliations
Contributions
All authors contributed equally.
Corresponding author
Additional information
Communicated by Tiago Pereira.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A. Proof of Theorem 3.1
Appendix A. Proof of Theorem 3.1
Proof of Theorem 3.1
First part: Let B be a closed, bounded, uniformly absorbing set for the system \(\{U_\sigma (t,s):t\geqslant s\}_{\sigma \in \Sigma }\) and consider the lifted set \(\mathbb {B} = B\times \Sigma \). We will prove that the skew-product semigroup in \(X\times \Xi \) has an exponential global attractor, and the existence of an exponential uniform attractor for \(\{U_\sigma (t,s):t\geqslant s\}\) will follow readily.
Then it is easy to see that if \(\{S(t): t\in {\mathbb R}\}\) is the skew-product semigroup generated by the system \(\{U_\sigma (t,s)\}_{\sigma \in \Sigma }\), then \(S(\tau ) \mathbb {B} \subset \mathbb {B}\).
For all \(n\in \mathbb N\), consider the covering of the finite-dimensional set \(\Sigma \) with a minimum number of open balls of \(\Xi \) of centers \(\{\Sigma _-^n, \dots , \sigma _N^n\}\) and radius \(R_n = R\nu ^n/P e^{\zeta n \tau } L(n\tau )\). We name these balls \(B_i^n = B_\Xi (\sigma _i^n, R_n)\). The number \(N = N_n\) depends on the box counting dimension of \(\Sigma \). More precisely, for every \(\delta > 0\), there exists \(n_0 = n_0(\delta )\) such that
Let \(\nu \in (0,1)\) and \(R> 0\), \(x_0 \in \mathcal {B}\) be such that \(\mathcal {B} = B_X(x_0, R) \cap \mathcal {B}\).
The following inductive argument is the famous smoothing technique that was used to prove finite dimensionality of attractors in many settings Cholewa et al. (2008); Carvalho and Sonner (2013); Efendiev et al. (2004). It consists of two steps. First we cover the unitary ball in Y with finitely many balls in X (we can do this because Y is compactly embedded in X). Then we apply the evolution process in these X balls and use the smoothing condition (\(H_2\)) to show that the image is bounded in Y. But since bounded sets in Y are precompact in X, we can start again and cover it with balls in X, keeping track of the number of balls needed as the radius decrease.
To start with, let \(\kappa = \kappa (\tau )\). Since Y is compactly embedded in X, the unit ball \(B_Y(0,1)\) in Y can be covered by \(N_{\frac{\nu }{2\kappa }}\) balls of radius \(\frac{\nu }{2\kappa }\) in X, that is:
Now, let \(\sigma \in \Sigma \) be arbitrary, it follows from \((H_{2})\) that:
Let \(y_\sigma = U_\sigma (\tau ,0)x_0\). Then it follows from (16), (36) and (37) that:
so there are \(q^\sigma _i \in \mathcal {B}\), \(i = 1, \cdots , N_{\frac{\nu }{2\kappa }}\) such that:
Suppose by induction that for some \(k \geqslant 1\), we have \(N_X(U_\sigma (k \tau ,0) \mathcal {B}, R\nu ^k) \leqslant N_{\frac{\nu }{2\kappa }}^k\) (we just proved that this is true for \(k = 1\)). Then, there exist \(q_i^\sigma \subset \mathcal {B}\), \(i = 1, \cdots , N_{\frac{\nu }{2\kappa }}^k\) such that:
Now, by the system of processes property:
Then, by (40),
and for each i, we have by the smoothing property \((H_{2})\) and self absorption of \(\mathcal {B}\) that:
for \(p_{i, j} \in \mathcal {B}\). Then, using (41) and (42), we deduce \(N_X(U_\sigma ((k+1) \tau ,0) \mathcal {B}, R\nu ^{k+1}) \leqslant N_{\frac{\nu }{2\kappa }}^{k+1}\).
Then, we proved by induction that for any \(\sigma \in \Sigma \),
For each \(\sigma \in \Sigma \), chose a set of centers \(W^n(\sigma ) \subset X\) such that \(\# W^n(\sigma ) \leqslant N^n_{\frac{\nu }{2\kappa }}\) and:
Now, for each \(n\in \mathbb N\) and \(j\in \{1, \dots N(n)\}\), we define:
Second part:
It follows from the definition of skew-product semigroup that if \(\Pi _1: X\times \Xi \rightarrow X\) is the projection onto the first coordinate, then:
Next we will show that:
Indeed, if \(y\in \Pi _1 S(n\tau ) \mathbb {B}\), then \(y = U_\sigma (n\tau ,0)x\), for some \(\sigma \in \Sigma \), \(x\in B\).
By the covering of \(\Sigma \), there exists \(i\in \{1, \dots , N(n)\}\) such that
And by the definition of the sets \(\mathcal {U}^i(n)\), we have:
Moreover, by \((H_5)\), we have:
And the estimates (46) and (47) imply (45). Notice that, more precisely, we showed that:
Now, for each \(v\in \mathcal {U}^i(n)\) take, if existing, \((u_v, \xi _v) \in \mathbb {B}\) such that \(\xi _v\in B_i^n\) and
If no such pair \((u_v, \xi _v)\) exists, we can remove the element v from \(\mathcal {U}^i(n)\) still preserving the estimate (48).
Now we define \(\mathbb {U}^i(n) = \{S(n\tau ) (u_v, \xi _v): v\in \mathcal {U}^i(n)\}\), and
Now, using (45) and the definition of \(\mathbb {U}(n)\), it is not hard to see that:
We also notice that \(\mathbb {U}(n) \subset S(n\tau ) \mathbb {B}\) and
Third part:
We define now the sets:
The set \(\mathbb {M}^d: = \overline{\bigcup _{k\in \mathbb {N}} \mathbb {E}(k)}\) will be the exponential attractor for the discrete semigroup \(\{S(n\tau ): n\in \mathbb N\}\). We will be able to explicitly calculate the cardinality of the sets \(\mathbb {E}(n)\), which will help us calculate the box-counting dimension of the global attractor \(\mathbb {M}^d\).
First, notice that \(S(\tau ) \mathbb {M}^d \subset \mathbb {M}^d\). Now if \(n\in \mathbb N\), we have:
For each \((x,\sigma ) \in \mathbb {B}\), there exists a \(i\in \{1, \dots , N(n)\}\) such that \(\sigma \in B_i^n\), and \(v\in \mathcal {U}^i(n)\) such that
and there exist \((u_v, \xi _v)\) such that \(\xi _v \in B^n_i\), and
And we have \(u_n = S(n\tau ) (u_v, \xi _v) \in \mathbb {U}(n)\) and:
which implies that
which gives the exponential attraction of \(\mathbb {B}\) by \(\mathbb {M}^d\) under the action of \(\{S(n\tau ): n\in \mathbb N\}\). Since \(\mathbb {B}\) is an absorbing set for the semigroup, the exponential attraction extends to every bounded set \(\mathbb {D} \subset \mathbb {X}\).
Fourth part: Now let us prove that \(\mathbb {M}^d\) has finite box-counting dimension. Since \(\mathbb {U}(n) \subset S(n\tau )\mathbb {B}\), it can be shown by induction that \(\mathbb {E}((n+j)\tau ) \subset S((n+j)\tau ) \mathbb {B} \subset S(n\tau ) \mathbb {B}\), which implies:
From the estimates on the cardinality of the sets \(\mathbb {U}(n)\), \(n\in \mathbb N\) and the construction of the sets \(\mathbb {E}(n)\), \(n\in \mathbb N\), we obtain:
It has been proved before that
Which implies that:
Then, we have:
Which implies that
To estimate this, we notice that
Finally, by definition of N(n), we have:
where we used the box-counting dimension \(d_\Sigma \) of \(\Sigma \) in \(\Xi \).
Then, we conclude that:
Fifth part: Now, we will extend the exponential attractor \(\mathbb {M}^d\) to an exponential attractor of the continuous semigroup \(\{S(t): t\geqslant 0\}\). We define:
The positive semi-invariance of \(\mathbb {M}\) by \(\{S(t):t\in {\mathbb R}\}\) follows from the positive semi-invariance of \(\mathbb {M}^d\) by \(\{S(n\tau ): n\in \mathbb N\}\).
For the exponential attraction, let \(t\geqslant 0\) and we write \(t = n\tau + s\) with \(s\in [0,\tau )\). Using \((H_3)\), \((H_4)\) and \((H_5)\) we can show that S(t) is \(\gamma \)-Hölder continuous in \(\mathcal {B} \times \Sigma \) with a constant we call K(t), and K(t) can be taken uniform \((=K)\) for \(t\in [0,\tau ]\). Then we have:
This implies that \(\mathbb {M}\) attracts any bounded set in \(\mathbb {X}\) exponentially under the action of \(\{S(t):t\in \mathbb {R}\}\).
Finally, we estimate the box-counting dimension of \(\mathbb {M}\). Notice that:
where \(\Phi : [0,\tau ] \times \mathbb {M}^d \rightarrow \mathbb {M}\) is given by \(\Phi (t,(x,\sigma )) = S(t) (x,\sigma )\). Since \(\Phi \) is \(\theta \)-Hölder continuous in time and \(\gamma \)-Hölder continuous in the \(\mathbb {X}\) variable, it follows that:
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Carvalho, A.N., Langa, J.A. & Moura, R.O. Finite Fractal Dimension of Uniform Attractors for Non-Autonomous Dynamical Systems with Infinite-Dimensional Symbol Space. J Nonlinear Sci 35, 70 (2025). https://doi.org/10.1007/s00332-025-10169-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00332-025-10169-0