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Radu Saghin

Publications and source records attributed to Radu Saghin.

13 recordsLinked to original sources
arXiv2025

Inverse SRB measures for endomorphisms on surfaces

We extend D. Burguet's construction of SRB measures for the non-invertible scenario obtaining hyperbolic invariant measures with absolutely continuous disintegrations on stable manifolds for a certain class of endomorphisms on the two-torus. The constructed measures also maximize the folding entropy, and we call them equidistributed inverse SRB measures. For dissipative perturbations of the examples given by M. Andersson, P. Carrasco and R. Saghin we obtain existence, uniqueness and continuity with respect to the map of the equidistributed inverse SRB measures as well as the SPR property of the stable geometric potential for such maps.

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arXiv2023

On invariant holonomies between centers

We prove that for $C^{1+\theta}$, $\theta$-bunched, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are $C^1$ and the derivative depends continuously on the points and on the map. Also for $C^{1+\theta}$, $\theta$-bunched partially hyperbolic diffeomorphism, the derivative cocycle restricted to the center bundle has invariant continuous holonomies which depend continuously on the map. This generalizes previous results by Pugh-Shub-Wilkinson, Burns-Wilkinson, Brown, Obata, Avila-Santamaria-Viana, Marin.

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arXiv2023

Shub's example revisited

For a class of robustly transitive diffeomorphisms on $\mathbb T^4$ introduced by Shub in [24], satisfying an additional bunching condition, we show that there exits a $C^2$ open and $C^r$ dense subset $\mathcal U^r$, $2\leq r\leq\infty$, such that any two hyperbolic points of $g\in \mathcal U^r$ with stable index $2$ are homoclinically related. As a consequence, every $g\in \mathcal U^r$ admits a unique homoclinic class associated to the hyperbolic periodic points with index $2$, and this homoclinic class coincides to the whole ambient manifold. Moreover, every $g\in \mathcal U^r$ admits at most one measure with maximal entropy, and every $g\in\mathcal U^{\infty}$ admits a unique measure of maximal entropy.

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arXiv2022

Non-uniformly hyperbolic endomorphisms

We show the existence of large $\mathcal C^1$ open sets of area preserving endomorphisms of the two-torus which have no dominated splitting and are non-uniformly hyperbolic, meaning that Lebesgue almost every point has a positive and a negative Lyapunov exponent. The integrated Lyapunov exponents vary continuously with the dynamics in the $\mathcal C^1$ topology and can be taken as far away from zero as desired. Explicit real analytic examples are obtained by deforming linear endomorphisms, including expanding ones. The technique works in nearly every homotopy class and the examples are stably ergodic (in fact Bernoulli), provided that the linear map has no eigenvalue of modulus one.

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arXiv2021

Extended flexibility of Lyapunov exponents for Anosov diffeomorphisms

Bochi-Katok-Rodriguez Hertz proposed in [BKH21] a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong hyperbolic properties we establish extended flexibility results for their Lyapunov exponents. We give examples of Anosov diffeomorphisms with the strong unstable exponent larger than the strong unstable exponent of the linear part. We also give examples of derived from Anosov diffeomorphisms with the metric entropy entropy larger than the entropy of the linear part. These results rely on a new type of deformation which goes beyond the previous Shub-Wilkinson and Baraviera-Bonatti techniques for conservative systems having some invariant directions. In order to estimate the Lyapunov exponents even after breaking the invariant bundles, we obtain an abstract result (Theorem 3.1) which gives bounds on exponents of some specific cocycles and which can be applied in various other settings. We also include various interesting comments in the appendices: our examples are $\mathcal{C}^2$ robust, the Lyapunov exponents are continuous (with respect to the map) even after breaking the invariant bundles, a similar construction can be obtained for the case of multiple eigenvalues.

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arXiv2020

Regularity of Lyapunov Exponents for Diffeomorphisms with Dominated Splitting

We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in $[0,1]$), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time one map of a geodesic flow of a surface of negative curvature the metric entropy of the volume is Lipschitz with respect to the parameter. At the time one map of a geodesic flow on a manifold of constant negative curvature the topological entropy is differentiable with respect to the parameter, and we give a formula for the derivative. Under some regularity conditions, the critical points of the Lyapunov exponent function are non-flat (the second derivative is nonzero for some families). Also, again under some regularity conditions, the criticality of the Lyapunov exponent function implies some rigidity of the map, in the sense that the volume decomposes as a product along the two complimentary foliations. In particular for area preserving Anosov diffeomorphisms, the only critical points are the maps smoothly conjugated to the linear map, corresponding to the global extrema.

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arXiv2019

Lyapunov exponents and rigidity of Anosov automorphisms and skew products

In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to $L$. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation $f$ of $f_0$ with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.

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arXiv2018

Entropy of partially hyperbolic flows with center dimension two

In this article we study the regularity of the topological and metric entropy of partially hyperbolic flows with two-dimensional center direction. We show that the topological entropy is upper semicontinuous with respect to the flow, and we give an example where the lower semicontinuity fails. We also show that if such a flow has no fixed points, then it is entropy expansive, and consequently the metric entropy function is upper semicontinuous, there exist equilibrium states (and measures of maximal entropy), and principal symbolic extensions.

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arXiv2012

Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms

We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.

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arXiv2011

Invariant measures for Cherry flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.

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arXiv2006

Geometric Expansions, Lyapunov Exponents and Foliations

We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an application, we show examples of systems with persistent non- absolute continuous center and weak unstable foliations. This generalizes the remarkable results of Shub and Wilkinson to cases where the center manifolds are not compact.

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arXiv2006

Topological Entropy and Partially Hyperbolic Diffeomorphisms

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique non-trivial homologies. We prove the following two results: if the center foliation is one dimensional, then the topological entropy is locally a constant; and if the center foliation is two dimensional, then the topological entropy is continuous on the set of all $C^\8$ diffeomorphisms. The proof uses a topological invariant we introduced; Yomdin's theorem on upper semi-continuity; Katok's theorem on lower semi-continuity for two dimensional systems and a refined Pesin-Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

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