Let $f$ be an endomorphism of $\mathbb{CP}^k$ and $ν$ be an $f$-invariant measure with positive Lyapunov exponents $(λ_1,\...,λ_k)$. We prove a lower bound for the pointwise dimension of $ν$ in terms of the degree of $f$, the exponents of $ν$ and the entropy of $ν$. In particular our result can be applied for the maximal entropy measure $μ$. When $k=2$, it implies that the Hausdorff dimension of $μ$ is estimated by $\dim_{\cal H} μ\geq {\log d \over λ_1} + {\log d \over λ_2}$, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the $ν$-generic inverse branches of $f^n$ in $\mathbb{CP}^k$. Our tools are a volume growth estimate for the bounded holomorphic polydiscs in $\mathbb{CP}^k$ and a normalization theorem for the $ν$-generic inverse branches of $f^n$.
We prove that the minimal chain recurrence classes of a holomorphic endomorphism of $\mathbb P^k$ have finitely many connected components. We also obtain results on arbitrary classes. These strong constraints on the topological dynamics in the phase space are all deduced from the associated action on a space of currents.
We study the structure and the Lyapunov exponents of the equilibrium measure of endomorphisms of $\mathbb P^k$ preserving a fibration. We extend the decomposition of the equilibrium measure obtained by Jonsson for polynomial skew products of $\mathbb C^2$. We also show that the sum of the sectional exponents satisfies a Bedford-Jonsson formula when the fibration is linear, and that this function is plurisubharmonic on families of fibered endomorphisms. In particular, the sectional part of the bifurcation current is a closed positive current on the parameter space.
For an algebraic family $(f_t)$ of regular quadratic polynomial endomorphisms of $\mathbb{C}^2$ parametrized by $\mathbb{D}^*$ and degenerating to a Hénon map at $t=0$, we study the continuous (and indeed harmonic) extendibility across $t=0$ of a potential of the bifurcation current on $\mathbb{D}^*$ with the explicit computation of the non-archimedean Lyapunov exponent associated to $(f_t)$. The individual Lyapunov exponents of $f_t$ are also investigated near $t=0$. Using $(f_t)$, we also see that any Hénon map is accumulated by the bifurcation locus in the space of quadratic holomorphic end…
We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and Ueda in the semiattracting setting. We also prove the discontinuity of the filled Julia set for such perturbations of regular polynomials.
In this paper, we discuss the equidistribution phenomena for holomorphic endomorphisms over $\mathbb{P}^k$ in the case of bidegree $(p,p)$ with $1<p<k$. We prove that if $f:\mathbb{P}^k\to\mathbb{P}^k$ is a holomorphic endomorphism of degree $d\geq 2$ and $T^p$ denotes the Green $(p,p)$-current associated with $f$, then there exists a proper invariant analytic subset $E$ for $f$ such that $d^{-pn}(f^n)^*(S)\to T^p$ exponentially fast in the current sense for every positive closed $(p,p)$-current $S$ of mass 1 such that $S$ is smooth on $E$.
We develop a new method, based on pluripotential theory, to study the transfer (Perron-Frobenius) operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a holomorphic endomorphism and a suitable continuous weight. This method allows us to prove the existence and uniqueness of the equilibrium state and conformal measure for very general weights (due to Denker-Przytycki-Urbański in dimension 1 and Urbański-Zdunik in higher dimensions, both in the case of H\''older continuous weights). We establish a number of properties of the equilibrium states, including mixing, K-mixing, mixing of all…
Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $μ$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\mathbb{P}^k,f,μ)$. Our class $\cal{U}$ of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map $ω: (Σ, s, ν) \to (\mathbb{P}^k,f,μ)$. We obtain the invariance principle for an observable $ψ$ on $(\mathbb{P}^k,f,μ)$ by applying Philipp-Stout's theorem for $χ= ψ\circ ω$ on $(Σ, s, ν)$. The invariance principle implies…
We study the dynamics of the family $f_c(x, y)= (xy+c, x)$ of endomorphisms of $\mathbb{R}^2$ and $\mathbb{C}^2$, where $c$ is a real or complex parameter. Such maps can be seen as perturbations of the map $f_0(x,y)=(xy,x)$, which is a complexification of the Anosov torus map $(u,v) \mapsto (u+v,u)$.
Let $f$ be an endomorphism of $\mathbb{C}\mathbb{P}^k$ of degree $>1$, and assume that for any cyclic Fatou component $W$ of $f$ having a period $p\in\mathbb{N}$, the equilibrium measure $μ_f$ has a positive charge on the boundary of $W$ if and only if $f^{-p}(W)=W$. Then we obtain a locally uniform a priori bound of the dynamics of $f$, which in particular yields a Diophantine-type estimate of the dynamics of $f$ on its domaines singuliers. We also point out that in the case of $k=1$, the statement of our assumption is related to both the impossibility for the Julia set of $f$ to be the bound…
In the context of holomorphic families of endomorphisms of $\mathbb P^k$, we prove that stability in the sense of [BBD18] is equivalent to a summability condition for the post-critical mass and to the convergence of a suitably defined ramification current. This allows us to both simplify the approach of [BBD18] and better relate stability to post-critical normality.
Let $f$ be an holomorphic endomorphism of $\mathbb{C}\mathbb{P}^k$. We construct by using coding techniques a class of ergodic measures as limits of non-uniform probability measures on preimages of points. We show that they have large metric entropy, close to $\log d^k$. We establish for them strong stochastic properties and prove the positivity of their Lyapunov exponents. Since they have large entropy, those measures are supported in the support of the maximal entropy measure of $f$. They in particular provide lower bounds for the Hausdorff dimension of the Julia set.
The goal of this paper is to study some basic properties of the Fatou and Julia sets for a family of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2$. We are particularly interested in studying these sets for semigroups generated by various classes of holomorphic endomorphisms of $\mathbb{C}^k,\; k \ge 2.$ We prove that if the Julia set of a semigroup $G$ which is generated by endomorphisms of maximal generic rank $k$ in $\mathbb{C}^k$ contains an isolated point, then $G$ must contain an element that is conjugate to an upper triangular automorphism of $\mathbb{C}^k.$ This generalizes a t…
We prove that a class of weakly partially hyperbolic endomorphisms on $\mathbb{T}^2$ are dynamically coherent and leaf conjugate to linear toral endomorphisms. Moreover, we give an example of a partially hyperbolic endomorphism on $\mathbb{T}^2$ which does not admit a centre foliation.
We establish the existence of a spectral gap for the transfer operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. The main issue to overcome is the rigidity of the complex objects, since the transfer operator is a non-holomorphic perturbation of the operator $f_*$. The system is moreover non-uniformly hyperbolic and one may have critical points on the Julia set. The construction of our norm requires the introduction and study of …
It is expected that a totally invariant divisor of a non-isomorphic endomorphism of the complex projective space is a union of hyperplanes. In this paper, we compute an upper bound for the degree of such a divisor. As a consequence, we prove the linearity of totally invariant divisors with isolated singularities.
We develop an algorithm that determines, for a given squarefree binary form $F$ with real coefficients, a smallest representative of its orbit under $\operatorname{SL}(2,\mathbb Z)$, either with respect to the Euclidean norm or with respect to the maximum norm of the coefficient vector. This is based on earlier work of Cremona and Stoll. We then generalize our approach so that it also applies to the problem of finding an integral representative of smallest height in the $\operatorname{PGL}(2,\mathbb Q)$ conjugacy class of an endomorphism of the projective line. Having a small model of such an …
We classify noninvertible, holomorphic selfmaps of the projective plane that preserve an algebraic web. In doing so, we obtain interesting examples of critically finite maps.
In this paper we prove the following theorem. Let $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$ be a dominate polynomial endomorphisms defined over an algebraically closed field $k$ of characteristic $0$. If there are no nonconstant rational function $g:\mathbb{A}^2-rightarrow \mathbb{P}^1$ satisfying $g\circ f=g$, then there exists a point $p\in \mathbb{A}^2(k)$ whose orbit under $f$ is Zariski dense in $\mathbb{A}^2$. This result gives us a positive answer to a conjecture of Amerik, Bogomolov and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.
Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $x\in J_2$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means that the super-saddle cycle is not contained in a critical co…
The problem of existence of standard (i.e. product-type) invariant MASAs for endomorphisms of the Cuntz algebra O_n is studied. In particular endomorphisms which preserve the canonical diagonal MASA D_n are investigated. Conditions on a unitary in O_n equivalent to the fact that the corresponding endomorphism preserves D_n are found, and it is shown that they may be satisfied by unitaries which do not normalize D_n. Unitaries giving rise to endomorphisms which leave all standard MASAs invariant and have identical actions on them are characterized. Finally some properties of examples of finite-…
We consider surjective endomorphisms f of degree > 1 on projective manifolds X of Picard number one and their f^{-1}-stable hypersurfaces V, and show that V is rationally chain connected. Also given is an optimal upper bound for the number of f^{-1}-stable prime divisors on (not necessarily smooth) projective varieties.
LLet $f$ be a holomorphic endomorphism of $\mathbb P^ 2$ of degree $d \geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $ν$. We infer several applications. The first one shows that the currents $S$ containing a measure of entropy $h\_ν> \log d$ have a directional dimension $>2$, which answers a question by de Thélin-Vigny. The second application asserts that the Dujardin's semi-extremal endomorphisms are close to suspensions of one-dimensional Lattès maps. Finally, we obtain an upper bound for the dimension of the eq…
Let $A_1, A_2\in \mathbb C(z)$ be rational functions of degree at least two that are neither Lattès maps nor conjugate to $z^{\pm n}$ or $\pm T_n.$ We describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of $(\mathbb P^1(\mathbb C))^2$ of the form $(z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).$ In particular, we show that if $A\in \mathbb C(z)$ is not a "generalized Lattès map", then any $(A,A)$-invariant curve has genus zero and can be parametrized by rational functions commuting with $A$. As an application, for $A$ defined over a subfield $K$ of $ \mathbb C$ we give a…
Let $S$ and $T$ be hyperbolic endomorphisms of $\mathbb{T}^d$ with the property that the span of the subspace contracted by $S$ along with the subspace contracted by $T$ is $\mathbb{R}^d$. We show that the Hausdorff dimension of the intersection of the set of points with equidistributing orbits under $S$ with the set of points with nondense orbit under $T$ is full. In the case that $S$ and $T$ are quasihyperbolic automorphisms, we prove that the Hausdorff dimension of the intersection is again full when we assume that $\mathbb{R}^d$ is spanned by the subspaces contracted by $S$ and $T$ along w…
For any algebraically closed field $K$ and any endomorphism $f$ of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of $f$ are the Möbius transformations that commute with $f$, and these form a finite subgroup of $\operatorname{PGL}_2(K)$. In the moduli space of complex dynamical systems, the locus of maps with nontrivial automorphisms has been studied in detail and there are techniques for constructing maps with prescribed automorphism groups that date back to Klein. We study the corresponding questions when $K$ is the algebraic closure $\bar{\mathbb{F}}_p$ of a finite field. We use …
Let $ν$ be a probability measure that is ergodic under the endomorphism $(\times p, \times p)$ of the torus $\mathbb{T}^2$, such that $\dim πμ< \dim μ$ for some non-principal projection $π$. We show that, if both $m\neq n$ are independent of $p$, the $(\times m, \times n)$ orbits of $ν$ typical points will equidistribute towards the Lebesgue measure. If $m>p$ then typically the $(\times m, \times p)$ orbits will equidistribute towards the product of the Lebesgue measure with the marginal of $μ$ on the $y$-axis. We also prove results in the same spirit for certain self similar measures $ν$. The…
In \cite{Ioana:vNsuperrigidity}, Ioana introduced three new invariants of type II$_1$ factors: the one-sided fundamental group, the endomorphism semigroup and the set of right-finite bimodules. In \cite{Ioana:vNsuperrigidity}, he does not provide many computations of these invariants. In particular, the question whether these invariants can be trivial is left open. We give an explicit example of a type II$_1$ factor for which all three invariants are trivial. More generally, for any countable left-cancellative semigroup $G$, we construct a type II$_1$ factor $M$ whose endomorphism semigroup is…
We consider surjective endomorphisms f of degree > 1 on the projective n-space with n = 3, and f^{-1}-stable hypersurfaces V. We show that V is a hyperplane (i.e., deg(V) = 1) but with four possible exceptions; it is conjectured that deg(V) = 1 for any n > 1.
There is a natural conjugation action on the set of endomorphism of $¶^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $¶^N$, denoted $\mathcal{M}_d^N$. We construct invariant functions on this moduli space coming from to set of multiplier matrices of the periodic points. The basic properties of these functions are demonstrated such as that they are in the ring of regular functions of $\mathcal{M}_d^N$, methods of computing them, as well as the existence of relations. The main part of the article examines to what extend these invariant…
We extend the results of arXiv:2206.08295v2 by showing that any homothety in $\mathbb T^2$ is homotopic to a non-uniformly hyperbolic ergodic area preserving map, provided that its degree is at least $5^2$. We also address other small topological degree cases not considered in the previous article. This proves the existence of a $\mathcal C^1$ open set of non-uniformly hyperbolic systems, that intersects essentially every homotopy classes in $\mathbb T^2$, where the Lyapunov exponents vary continuously.
We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor.As a consequence under a mild assumption we prove exponential large deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.
Let $f$ be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or stabilizer group, of a given $f$ for this action is known to be a finite group. We determine explicit families that parameterize all endomorphisms defined over $\bar{\mathbb{Q}}$ of degree $3$ and degree $4$ that have a nontrivial automorphism, the \textit{automorphism locus} of the moduli space of dynamical systems. We analyze the geometry of these loci in the appropriate moduli space of dynamical sys…
We show that for a large class of potential functions and big coupling constant $λ$ the Schrödinger cocycle over the expanding map $x\mapsto bx ~( \text{mod} 1)$ on $\mathbb{T}$ has a Lyapunov exponent $>(\logλ)/4$ for all energies, provided that the integer $b\geq λ^3$.
Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $ϕ: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$ of End($M$) such that, if $f\in\mathcal{A}$, there exists a $f$ invariant measure $μ_{\max}$ supported on a periodic orbit that maximizes the integral of $ϕ$ among all $f$ invariant Borel probability measures.
Despite the invertible setting, Anosov endomorphisms may have infinitely many unstable directions. Here we prove, under transitivity assumption, that an Anosov endomorphism on a closed manifold $M,$ is either special (that is, every $x \in M$ has only one unstable direction) or for a typical point in $M$ there are infinitely many unstable directions. Other result of this work is the semi rigidity of the unstable Lyapunov exponent of a $C^{1+α}$ codimension one Anosov endomorphism and $C^1$ close to a linear endomorphism of $\mathbb{T}^n$ for $(n \geq 2).$ In the appendix we give a proof for er…
Given an integer nonsingular $n\times n$ matrix $M$ and a point $y \in \mathbb{R}^n/\mathbb{Z}^n$, consider the set $\tilde E(M,y)$ of vectors $x\in \mathbb{R}^n$ such that $y$ is not a limit point of the sequence $\{M^k x \mod \mathbb{Z}^n: k\in\mathbb{N}\}$. S.G. Dani showed in 1988 that whenever $M$ is semisimple and $y \in \mathbb{Q}^n/\mathbb{Z}^n$, the set $\tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary $y \in \mathbb{R}^n/\mathbb{Z}^n$ and integer nonsingular $M$, and in fact replacing the sequence of powers of $M$ by any …
For N d-actions by algebraic endomorphisms on compact abelian groups, the existence of non-mixing configurations is related to "S-unit type" equations and plays a role in limit theorems for such actions. We consider a family of endomorphisms on shift-invariant subgroups of F Z d p and show that there are few solutions of the corresponding equations. This implies the validity of the Central Limit Theorem for different methods of summation.
